Introduction
One-relator presentations impose a single equation on a free group, but their geometric behavior ranges from free groups to groups with distorted cyclic subgroups. Hyperbolicity rules out the latter behavior. The purpose of this paper is to show that it also forces a strong finite-cover geometry.
Theorem 1.1. Let X X X r ∈ F ( X ) r \in F(X) r ∈ F ( X )
G = F ( X ) / ⟨ ⟨ r ⟩ ⟩ G = F(X)/\langle\langle r\rangle\rangle G = F ( X ) / ⟨⟨ r ⟩⟩ is word-hyperbolic. Then G G G H ≤ G H \le G H ≤ G C C C Λ \Lambda Λ
π 1 ( C ) ≅ H and C ⟶ S Λ \pi_1(C) \cong H \quad\text{and} \quad C \longrightarrow S_{\Lambda} π 1 ( C ) ≅ H and C ⟶ S Λ is a cubical local isometry to the Salvetti complex of the right-angled Artin group defined by Λ \Lambda Λ
The compactness assertion is part of the conclusion. An abstract embedding of a finite-index subgroup in a right-angled Artin group would not by itself supply the model required in Theorem 1.1.
The inductive study of one-relator groups begins with Magnus’s Freiheitssatz: omitting a generator that occurs in a cyclically reduced relator leaves a subgroup freely generated by the remaining generators [44 ] . Magnus subsequently solved the word problem for one-relator groups [45 ] . These results established the importance of reducing a one-relator problem to simpler presentations while retaining control of distinguished free subgroups. The Magnus–Moldavanskii hierarchy develops this approach through splittings over those subgroups [39 ] , Sections 1.1–1.3. The geometric question is whether the splittings have the quasiconvexity needed to construct finite special covers.
A second line of inquiry concerns finite quotients. A group is residually finite if each nonidentity element survives in some finite quotient. Baumslag’s 1967 Conjecture A asks for this property for one-relator groups with nontrivial torsion [6 ] . His paper contrasts this question with non-Hopfian one-relator examples, which cannot be residually finite. Wise’s quasiconvex hierarchy theory proved the stronger virtual compact specialness conclusion for one-relator groups with torsion [66 ] . Theorem 1.1 extends the compact-special conclusion, and hence residual finiteness, to every hyperbolic one-relator group.
The cubical approach explains why this strengthening matters. Haglund–Wise’s special cube complexes connect nonpositive curvature with embeddings in right-angled Artin groups and, in the hyperbolic case, separation of quasiconvex subgroups in finite quotients [28 ] . Wise’s hierarchy theorem makes quasiconvex splittings a way to obtain this structure [66 ] ; Agol’s theorem supplies it for hyperbolic groups acting properly and cocompactly on CAT(0) cube complexes [2 ] . Thus the geometric conclusion controls both linear representations and finite-quotient information. Section 18 explains these consequences, including conjugacy separation and virtual retractions, as well as the fibering conclusions below.
In the torsion-free direction, Linton proved hyperbolicity and virtual specialness when all two-generator subgroups of a one-relator group are free [39 ] . His subsequent work identifies primitive extension groups as the terminal cases in the general hyperbolic problem and specifies the quasiconvex towers through which a terminal result propagates [40 ] . The remaining task is therefore a geometric construction for those primitive groups. Here we construct virtually compact special quotients, recover that structure using selective covers of the fillings, and then propagate it up the truncated tower. Wise singled out hyperbolic one-relator groups in his broader question about cocompact cubulation of hyperbolic groups with hierarchies [65 ] . Theorem 1.1 establishes the compact-special conclusion for this one-relator class.
We use free-by-cyclic in the subgroup-closed sense: a group H H H
1 ⟶ K ⟶ H ⟶ Q ⟶ 1 1 \longrightarrow K \longrightarrow H \longrightarrow Q \longrightarrow1 1 ⟶ K ⟶ H ⟶ Q ⟶ 1 with K K K Q ≤ Z Q \le\mathbb{Z} Q ≤ Z [37 ] . Their Corollary 1.3 identifies virtual compact specialness as a sufficient additional hypothesis. Combining that corollary with Theorem 1.1 gives a positive answer. Corollary 18.1 also records the Baumslag–Solitar-free consequence from the companion hyperbolicity theorem and a stronger finite-index embedding into a free-by-Z \mathbb{Z} Z
In the proof we use the hyperbolicity and Magnus-subgroup results of [52 ] at two specific points: embedded primitive extension groups are themselves hyperbolic, and their actual Magnus rows have the required hyperbolic geometry. The criterion in that paper excludes every Baumslag–Solitar subgroup B S ( m , n ) BS(m,n) B S ( m , n ) m n ≠ 0 mn \ne0 mn = 0
The recovery step is a filling criterion proved in Section 3. Its input is a coherent, locally indicable hyperbolic group of cohomological dimension at most two, with a finite classifying complex acyclic over its Hughes-free division field, and arbitrarily deep virtually compact special fillings along specified peripherals. We replace these by selective fillings that retain any prescribed finite collection of kernel factors. Their covers have a quasiconvex cubical hierarchy. This geometric control realizes the finite coefficient data of a chain contraction in ordinary Novikov completions for both signs of one character. The resulting finite-index subgroup maps onto Z \mathbb{Z} Z finite-rank free kernel, and hence is virtually compact special.
Before constructing the primitive fillings, we use this criterion to prove a second result. It will also supply the specialness of the periodic end groups encountered in that construction.
Theorem 1.2. Let F F F finitely generated free group and let ϕ : F → F \phi:F \to F ϕ : F → F ascending mapping torus
F ∗ ϕ = ⟨ F , t ∣ t − 1 f t = ϕ ( f ) , f ∈ F ⟩ F*_{\phi} = \langle F,t \mid t^{-1}ft = \phi(f),\ f \in F \rangle F ∗ ϕ = ⟨ F , t ∣ t − 1 f t = ϕ ( f ) , f ∈ F ⟩ is word-hyperbolic, then it is virtually compact special.
Hagen and Wise cubulate hyperbolic free-by-cyclic groups with finite-rank free kernel [27 ] , Theorem A]. Their irreducible theorem also covers expanding, fundamental-group-injective train-track maps whose transition matrices are irreducible and whose mapping-torus groups are hyperbolic [26 ] , Theorem B]. They posed the cubulation problem for hyperbolic ascending HNN extensions defined by general monomorphisms in [27 ] , Problem C]. Mutanguha constructs expanding relative immersions for injective nonsurjective endomorphisms and proves the Baumslag–Solitar criterion for arbitrary injective ascending tori [51 ] , Theorems 4.5 and 6.7]. He also records Kielak’s question whether hyperbolic ascending tori of injective nonsurjective free-group endomorphisms are virtually free-by-cyclic [51 ] , Overview, p. 3, Question (Virtual fibering)]. The proof below retains arbitrary injective input and establishes the compact-special conclusion. Linton proved that every hyperbolic free-by-cyclic group is virtually compact special, allowing its free kernel to have infinite rank [41 ] , Corollary 1.5]. The canonical kernel of a strict ascending torus is presented here as an increasing union of free groups; freeness of that union is not an input to our argument.
After Theorem 1.2 is proved, the ascending group’s field acyclicity and virtual RFRS property give a stronger fibering conclusion: it has a finite-index subgroup mapping onto Z \mathbb{Z} Z
Kielak’s virtual-fibering method realizes division-field coefficients in ordinary Novikov completions along an RFRS filtration [36 ] , Section 4.2 and Theorem 5.2]. Fisher’s later work relates this method to cohomological dimension of kernels [23 ] . Here the original group is not yet known to be virtually RFRS: the selective switch supplies the required finite coefficient data before the two-sided Novikov argument. Once compact specialness is established, Kielak–Linton’s theorems give the general one-relator fibering and ambient-embedding consequences [37 ] .
The remaining construction provides the primitive group’s good fillings. Its Magnus splitting has a height coordinate. We call the vertex and edge spaces at specified heights its rows ; these are the actual spaces in the developed Bass–Serre tree, not merely their quotient labels. Long paths with noncyclic stabilizers obstruct an acylindrical hierarchy. Section 5 controls them by a joint injection into one finite-dimensional homology space. Keeping this injection under subgroup restriction, rather than only its rank inequality, distinguishes coincident contributions from different path orbits.
Expanding trajectories then identify periodic ends and join them along bilateral trajectories (Section 8). Homology prevents circuits and rank growth in the resulting incidence trees. We close the corresponding quasiconvex groups under cyclic intersections, fold rows, and enlarge shared ends until they form a malnormal peripheral system. There are two termination arguments: horizontal joins with no homological growth stabilize because their increasing union is finitely generated, whereas each genuine enlargement at a shared line end strictly increases the injected homology dimension. The coefficient constructions in Sections 6 and 7 preserve the joint injections through these operations. The first retains prescribed subgroup supports in scalar chain equations; the second localizes semilinear affiliated coefficients to finitely many periodic cosets.
We fill finite-index subgroups of the peripheral height kernels , retaining the height direction. Exact intersection transfer (Section 14) makes the resulting splitting acylindrical. The compatible kernels of Section 15 also give exact link covers for the filled rows. Binary labels on these links have sparse changes; the changes specify horizontal wall attachments. Within each filled vertex row, panels join through midpoint slots in the central pieces and then through satellite and outer fans. This produces quasiconvex trees carrying binary cocycles (Sections 16 and 17). Keeping central and satellite panels separate lets the labels control departures introduced by later attachments, including at root extensions.
Three boundary configurations require different panel tests. The resulting separators satisfy Bergeron–Wise’s boundary criterion [7 ] ; Agol’s theorem then gives virtual compact specialness of the filled rows. The transferred hierarchy assembles the good quotients in Theorem 17.19. Finally Theorem 17.20 applies the filling criterion to the unfilled primitive group, and the truncated one-relator tower proves Theorem 1.1.
Background, conventions, and the reduction
We write VCS for virtually compact special , QC for quasiconvex, and fg for finitely generated. RFRS means residually finite rationally solvable. An ascending mapping torus always uses an injective endomorphism unless explicitly described as a presentation torus before passing to its stable image. All subgroup occurrences in a Bass–Serre tree include their actual conjugating frames. Thus two occurrences of isomorphic subgroups need not be the same subgroup or the same peripheral object.
Cubulation and hierarchy results
We use the following standard results in their compact forms.
Proposition 2.1 (Cubulation and quasiconvex hierarchies).
A word-hyperbolic group acting properly and cocompactly on a CAT(0) cube complex is virtually compact special.
A hyperbolic group with a finite graph-of-groups decomposition whose edge groups are quasiconvex in the ambient group and whose vertex groups are hyperbolic VCS is VCS. The same applies to a finite hierarchy of such decompositions.
A compact special cube complex admits a cubical local isometry to a Salvetti complex. Its defining graph can be taken finite.
These are the virtual special theorem [2 ] , the quasiconvex hierarchy theorem [66 ] , and the special-complex characterization [28 ] , respectively. For the last assertion, only finitely many hyperplanes and crossing relations occur in a finite complex. Passing to a further special cover, when needed, provides the oriented two-sided convention.
The hierarchy statement with VCS terminal groups follows also by cutting a compact special cover of each terminal model along its finitely many two-sided hyperplane families. The cutting traces are convex, hence quasiconvex, and the resulting hierarchy ends in trivial groups. Finite-index supergroups are allowed in the hyperbolic hierarchy class. In any finite splitting used here, quasiconvexity of an edge group is asserted in the ambient group, not merely in an adjacent vertex group. Finitely generated vertex groups are then undistorted: in a tree of Cayley spaces, replace excursions from a target vertex through incident edge cosets by paths in those cosets of comparable length.
We use residual finiteness, virtual RFRS, quasiconvex subgroup separability and virtual retractions for hyperbolic VCS groups [1 , 28 , 29 , 66 ] . The malnormal special quotient theorem supplies designated finite-index normal subgroups P ˙ i ⊴ P i \dot{P}_i \trianglelefteq P_i P ˙ i ⊴ P i N ˙ i ⊴ P ˙ i \dot{N}_i \trianglelefteq\dot{P}_i N ˙ i ⊴ P ˙ i P ˙ i \dot{P}_i P ˙ i [3 ] . We retain this containment condition at every use; it is not replaced by finite avoidance. Additional finite avoidance conditions can be imposed by further intersections. Exact peripheral embeddings follow from the relative Dehn filling theorem [55 ] .
For relative hyperbolicity we use peripheralization of finite almost malnormal quasiconvex families and the uniform relative Dehn filling theorem [11 , 53 , 55 ] . For rows that are only relatively finitely generated, the fine-graph to relative-presentation comparison is [24 ] ; its hypotheses and exact infinite vertex stabilizers are checked at the constructions below. In particular, bounded isolated peripheral components in sufficiently long fillings have labels from a fixed finite list in the original peripheral groups. Later arguments explain the additional tests needed for exact subgroup-intersection transfer; that stronger conclusion is not assumed as part of the ordinary filling theorem.
The primitive extension reduction
Here is the shape of the groups that remain after the reduction. For k ≥ 1 k \ge1 k ≥ 1 a i = t − i a t i a_i = t^{-i}at^i a i = t − i a t i
⟨ a , t ∣ pr p / q ( x , y ) ⟩ , ⟨ a , t ∣ pr p / q ( x y , z ) ⟩ , \langle a,t \mid\operatorname{pr}_{p/q}(x,y)\rangle,\qquad\langle a,t \mid\operatorname{pr}_{p/q}(xy,z)\rangle, ⟨ a , t ∣ pr p / q ( x , y )⟩ , ⟨ a , t ∣ pr p / q ( x y , z )⟩ , where p , q p,q p , q pr p / q ( v , w ) \operatorname{pr}_{p/q}(v,w) pr p / q ( v , w ) v , w v,w v , w p / q p/q p / q x , y x,y x , y a 0 , … , a k − 1 a_0,\ldots,a_{k-1} a 0 , … , a k − 1 a 1 , … , a k a_1,\ldots,a_k a 1 , … , a k F ( a 1 , … , a k − 1 ) F(a_1,\ldots,a_{k-1}) F ( a 1 , … , a k − 1 ) z z z
Proposition 2.2 (Reduction). To prove Theorem 1.1, it suffices to prove that every torsion-free hyperbolic primitive extension group in the two forms of [40 ] , Section 5] is VCS.
Proof. The trivial relator gives a free group. Wise’s virtual compact specialness theorem handles the torsion case [66 ] . For a torsion-free group, use the maximal factored one-relator tower and primitive extension reduction of [40 ] . The primitive terminal groups have finite one-relator presentations and embed in G G G
A hyperbolic group contains no BS ( m , n ) \operatorname{BS}(m,n) BS ( m , n ) m n ≠ 0 mn \ne0 mn = 0 [52 ] makes each of them hyperbolic; no general assertion that finitely generated subgroups of hyperbolic groups are hyperbolic is being made.
With these terminal groups hyperbolic, the tower truncated at its primitive extension complex has the required acylindricity and quasiconvex inclusions by [40 ] . Only this truncated tower is used; no acylindricity is asserted for the unused remainder of the maximal tower below it. If each primitive terminal group is VCS, Proposition 2.1 propagates VCS through these upward steps. If the tower reaches an elementary terminal group without a primitive obstruction, the same argument starts there. This proves the reduction with the compact conclusion.
The precise primitive presentations and their actual refined rows are given in Section 5. They have two generators and one nontrivial relator, so their standard aspherical complexes have Euler characteristic zero. Torsion-free one-relator groups are locally indicable [13 , 30 ] ; see also the direct proof in [31 ] . They are coherent [34 ] . For the nontrivial non-power presentations here, Lyndon’s asphericity theorem [43 ] is also supplied by [5 ] ; specialize to the free group on its Cayley tree and a singleton non-power relator. The root quotient is trivial, so the resulting exact augmented sequence is the cellular sequence of the simply connected presentation cover; that cover is acyclic and hence contractible. The rational Linnell division fields are available, with Hughes-free subgroup compatibility, by [35 , 33 ] .
For such a primitive group, the universal-cover chain complex is acyclic over its division field. The Fox 2-boundary is a nonzero column, hence injective over the field; degree-zero homology vanishes because the group is nontrivial, and Euler characteristic zero then gives degree-one acyclicity. These are the finiteness and field hypotheses used by the filling criterion.
Coefficient and tree conventions
For a subgroup U U U D U D_U D U U ( U ) \mathcal{U}(U) U ( U )
d ( r s ) = d ( r ) + r d ( s ) , ∂ d ( r ) = r − 1. d(rs) = d(r) + r d(s), \qquad\partial d(r) = r - 1. d ( r s ) = d ( r ) + r d ( s ) , ∂ d ( r ) = r − 1. Changing a subgroup frame by s s s s − 1 s^{-1} s − 1 H ∗ ( V ; D U ) H_\ast(V;D_U) H ∗ ( V ; D U ) V ≤ U V \le U V ≤ U [59 , 14 ] .
All matrix traces in the affiliated-operator argument are compatible unnormalized extensions of the group trace. This ensures equality of traces for rectangular products in the two possible orders. Matrix dimensions remain finite, so this convention does not change tightness in the measure topology.
Proof of Theorem 1.1, assuming the later constructions. Theorem 17.20 proves the primitive case required by Proposition 2.2. Apply that proposition and then the Salvetti characterization in Proposition 2.1. This gives a finite connected special cover, its finite-index fundamental group, and the stated local isometry. Theorem 1.2 is established in Section 4.
A filling criterion and progressive characters
This section separates the geometric and algebraic ingredients of the filling criterion. The geometric ingredient preserves a prescribed finite part of a filling kernel in a virtually compact special quotient of a finite-index subgroup. The algebraic ingredient uses these quotients to resolve the coefficient problems left by an FRRS chain.
We specify the division-ring hypotheses, rather than assume that local indicability alone supplies them. For a group J J J D ( J ) D(J) D ( J ) Q J \mathbb{Q}J Q J N = ker χ N = \ker\chi N = ker χ χ \chi χ J J J
D ( J ) = Ore ( D ( N ) ∗ ( J / N ) ) . (1) D(J) = \operatorname{Ore}(D(N) \ast(J/N)). \tag*{(1)} D ( J ) = Ore ( D ( N ) ∗ ( J / N )) . (1) Here the crossed product is embedded using coset representatives. Its quotient J / N J/N J / N [32 , 38 ] for their origins and [35 ] and [33 ] for the locally indicable affiliated realization and the normal-subgroup crossed-product assertions. Arbitrary subgroup coset independence is justified separately at the start of Section 6. We refer to this collection of properties as the subgroup-field hypotheses .
Theorem 3.1 (Filling criterion). Let G ∗ G_\ast G ∗
(i) G ∗ G_\ast G ∗
(ii) G ∗ G_\ast G ∗ D ( G ∗ ) D(G_\ast) D ( G ∗ ) subgroup-field hypotheses , and the chains of a finite classifying space become acyclic over D ( G ∗ ) D(G_\ast) D ( G ∗ )
(iii) There is a finite malnormal family { D j } \{D_j\} { D j }
G ∗ ⟶ Q ∗ G_\ast\longrightarrow Q_\ast G ∗ ⟶ Q ∗ are hyperbolic and virtually compact special, and their inherited peripheral images are hyperbolic and virtually compact special.
In (iii), arbitrarily deep means that the peripheral kernels can avoid any prescribed finite subset of ⋃ j ( D j ∖ { 1 } ) \bigcup_j(D_j \setminus\{1\}) ⋃ j ( D j ∖ { 1 })
Then a finite-index subgroup of G ∗ G_\ast G ∗ Z \mathbb{Z} Z G ∗ G_\ast G ∗
The last conclusion uses cubulation of hyperbolic free-by-cyclic groups [27 ] and Agol’s theorem [2 ] . We prove the other conclusions below. In particular, the original peripheral images in Theorem 3.1 need not be finite.
Cofinite fillings and selective quotients
Discard trivial members of the peripheral family. If no members remain, the good filling is the identity, so hypothesis (iii) already says that G ∗ G_\ast G ∗ G ∗ G_\ast G ∗ Z \mathbb{Z} Z
By malnormal quasiconvex peripheralization, the pair ( G ∗ , { D j } ) (G_\ast,\{D_j\}) ( G ∗ , { D j }) [11 , 53 ] . Suppose a sufficiently deep good filling as in Theorem 3.1(iii) has been chosen, written q : G ∗ ⟶ Q q:G_\ast\longrightarrow Q q : G ∗ ⟶ Q P j = q ( D j ) P_j=q(D_j) P j = q ( D j ) Q Q Q P ˙ j ⊴ P j \dot{P}_j \trianglelefteq P_j P ˙ j ⊴ P j [3 ] . Each P j P_j P j L j ⊴ P j L_j \trianglelefteq P_j L j ⊴ P j M j = P ˙ j ∩ L j M_j=\dot{P}_j\cap L_j M j = P ˙ j ∩ L j R j = ( q ∣ D j ) − 1 ( M j ) R_j=(q|_{D_j})^{-1}(M_j) R j = ( q ∣ D j ) − 1 ( M j ) M j M_j M j R j R_j R j P j P_j P j M j = 1 M_j=1 M j = 1
K = ⟨ ⟨ R j : j ⟩ ⟩ G ∗ , Q ∗ = G ∗ / K , D ‾ j = D j / R j . (2) K=\langle\langle R_j:j\rangle\rangle_{G_*},\qquad Q_*=G_*/K,\qquad\overline{D}_j=D_j/R_j. \tag*{(2)} K = ⟨⟨ R j : j ⟩ ⟩ G ∗ , Q ∗ = G ∗ / K , D j = D j / R j . (2) Every depth requirement below can be imposed before this choice.
For sufficiently deep fillings, the Cohen–Lyndon theorem [63 ] identifies K K K
K = ∗ a ∈ A R a (3) K=\mathop{*}_{a\in\mathcal{A}}R_a \tag*{(3)} K = ∗ a ∈ A R a (3) where A = ∐ j Q ∗ / D ‾ j \mathcal{A}=\coprod_j Q_*/\overline{D}_j A = ∐ j Q ∗ / D j R a R_a R a R j R_j R j G ∗ G_* G ∗ G ∗ / D j G_*/D_j G ∗ / D j Q ∗ / D ‾ j Q_*/\overline{D}_j Q ∗ / D j K K K K K K K ≤ H K\leq H K ≤ H H H H H H H (3) .
Lemma 3.2 (Uniform selective hyperbolicity). The depth in (2) can be chosen so that the following holds for every finite-index preimage H ≤ G ∗ H\leq G_* H ≤ G ∗ K K K H H H K − ◃ H K_-\triangleleft H K − ◃ H H / K − H/K_- H / K − H H H
Proof. Use a proper Cayley space of G ∗ G_* G ∗
Coulon’s geometric small-cancellation theorem [17 ] applies to an invariant family of such hulls when every nonidentity rotation has sufficiently large infimal translation length. Its constants depend on the hyperbolicity and overlap bounds, not on the number of orbits. Rescale the original space first to meet those bounds, fix the cone radius, and then make the filling deep enough to meet the translation-length bound.
This last requirement is finite avoidance. Elements of bounded infimal displacement have representatives in finitely many conjugacy classes in a proper Cayley space. A nontrivial such class meets a malnormal peripheral group in at most one peripheral conjugacy class. Normality of R j R_j R j D j D_j D j
Now restrict the action to H H H H H H R j R_j R j D j D_j D j H / K − H/K_- H / K − H / K − H/K_- H / K −
Proposition 3.3 (Selective-cover quotient). Choose (2) sufficiently deep. Let H g i v e n H_{\mathrm{given}} H given G ∗ G_{\ast} G ∗ B ⊂ A \mathcal{B} \subset\mathcal{A} B ⊂ A H ≤ H g i v e n H \le H_{\mathrm{given}} H ≤ H given H g i v e n H_{\mathrm{given}} H given H H H B \mathcal{B} B
P = H / K − (4) P=H/K_{-} \tag*{(4)} P = H / K − (4) is hyperbolic and virtually compact special. The natural map to P P P ∗ a ∈ B R a \ast_{a\in\mathcal{B}}R_a ∗ a ∈ B R a
The injectivity follows at once from (3) , since the retained index set is H H H K ⊂ H K\subset H K ⊂ H P P P
Removing walls near the retained apices
Let X X X G ∗ G_{\ast} G ∗
Z = X / K , X ~ = X / K − , π : X ~ ⟶ Z , B = K / K − . (5) Z=X/K,\qquad\widetilde{X}=X/K_{-},\qquad\pi:\widetilde{X}\longrightarrow Z,\qquad B=K/K_{-}. \tag*{(5)} Z = X / K , X = X / K − , π : X ⟶ Z , B = K / K − . (5) The complex Z Z Z Q ∗ Q_{\ast} Q ∗ X X X X X X
The map π \pi π B B B
Pass first to a torsion-free finite-index subgroup V ≤ Q ∗ V\le Q_{\ast} V ≤ Q ∗ C C C V V V
f : Z ⟶ C . f:Z\longrightarrow C. f : Z ⟶ C . For example, define it on finitely many cell representatives and extend equivariantly, using contractibility of C C C
We record a control observation used repeatedly below. For any fixed radius, loops in a ball of that radius in Z Z Z Z Z Z C C C Z Z Z f f f (3.6) , not on the diameter or duration of the homotopy.
Fix a large radius R R R W 0 \mathcal{W}_0 W 0 R R R f ( B ) f(\mathcal{B}) f ( B ) m = ∣ W 0 ∣ m = |\mathcal{W}_0| m = ∣ W 0 ∣ H ‾ ≤ V ∩ ( H g i v e n / K ) \overline{H} \le V \cap(H_{\mathrm{given}}/K) H ≤ V ∩ ( H given / K ) H ‾ \overline{H} H W ∈ W 0 W \in\mathcal{W}_0 W ∈ W 0 2 m + dim C 2m + \dim C 2 m + dim C R R R
Here is why such a subgroup exists. Hyperplane stabilizers are quasiconvex and separable, and act cocompactly on their hyperplanes. For a fixed distance bound only finitely many double cosets of a hyperplane stabilizer can give translates at that distance. Separability excludes representatives of the double cosets not equal to the stabilizer itself, in a finite-index subgroup containing the stabilizer. Intersect these finitely many subgroups, and take a normal core in H g i v e n / K H_{\mathrm{given}}/K H given / K H H H
Delete the hyperplanes in H ‾ W 0 \overline{H}\mathcal{W}_0 H W 0 Y Y Y
p : C ⟶ Y . p : C \longrightarrow Y. p : C ⟶ Y . It is useful to include the elementary details of this projection.
Lemma 3.4 (Bounded wall collapse). The cubing Y Y Y p p p Y Y Y C C C v ∈ Y v \in Y v ∈ Y E v ⊂ C E_v \subset C E v ⊂ C σ ⊂ Y \sigma\subset Y σ ⊂ Y
E σ = ⋂ v ∈ σ ( 0 ) E v . E_\sigma= \bigcap_{v \in\sigma^{(0)}} E_v. E σ = v ∈ σ ( 0 ) ⋂ E v . These sets are nonempty bounded convex sets. A point mapping into an open cube σ \sigma σ dim C \dim C dim C E σ E_\sigma E σ R R R R R R R R R E σ E_\sigma E σ dim σ > 0 \dim\sigma> 0 dim σ > 0 R R R
Proof. Vertices of Y Y Y C C C
A combinatorial geodesic cannot cross more than m m m W 0 \mathcal{W}_0 W 0 m m m m m m Y Y Y
For a cube of Y Y Y C C C σ \sigma σ
No retained hyperplane comes within R R R B B B E v E_v E v E σ E_\sigma E σ dim σ > 0 \dim\sigma> 0 dim σ > 0
Sheets and a cubical branched model
We give the sheet construction with uniform collars. This avoids any assumption that the branch geometry is locally a cubical vertex star. Choose constants
r 0 < r < D < R r_0 < r < D < R r 0 < r < D < R in this order. The coarse return-of-homotopies observation following (3.6) has a fixed error at the chosen mesh scale. Choose r 0 r_0 r 0 dim C \dim C dim C r r r D D D 2 r + 10 r 0 + dim C 2r + 10r_0 + \dim C 2 r + 10 r 0 + dim C R R R H H H m m m
A near point for E σ E_\sigma E σ z ∈ X ~ z \in\widetilde{X} z ∈ X d ( f π ( z ) , E σ ) ≤ r 0 d(f\pi(z), E_\sigma) \le r_0 d ( f π ( z ) , E σ ) ≤ r 0 sheet if they are joined by a path whose f π f\pi f π r r r E σ E_\sigma E σ
Lemma 3.5 (Sheet stability and stabilizers). With the choices above, the following hold.
(i) Enlarging the path tolerance from r r r D D D
(ii) B B B E σ E_\sigma E σ dim σ > 0 \dim\sigma> 0 dim σ > 0
(iii) The stabilizer of a vertex sheet is generated by the stabilizers of the branch lifts accessible in that sheet. These are whole conjugates of retained factors in (3) . There are finitely many orbits of such accessible branch lifts under the sheet stabilizer.
Proof. We first prove the collar assertion. Project an excursion of a path towards the convex set E = E σ E = E_\sigma E = E σ C C C
The homotopy of an exterior point to its nearest point stays outside the interior of E E E E E E E E E Z Z Z (3.6) . Its uniformly bounded additional errors still avoid the branch images because R R R r r r E σ ⊂ E τ E_\sigma\subset E_\tau E σ ⊂ E τ τ ⊂ σ \tau\subset\sigma τ ⊂ σ
Paths within a fixed neighborhood of E E E E E E
If E E E E E E
For a positive-dimensional σ \sigma σ B B B E σ ⊂ E τ E_\sigma\subset E_\tau E σ ⊂ E τ B B B B B B
For a vertex chamber, let a deck transformation stabilize a sheet. Choose a path from a reference point to its translate within the allowed collar. Replace passages through branch lifts by link paths. Such paths exist because the links are connected; even a long path in a lifted link has image in the bounded star of its branch image. Contract the projected loop in a controlled enlargement of E E E
Lift this factorization. The discrepancy of a link loop is an element of the full stabilizer of the corresponding accessible branch lift. All the paths to these lifts stay in the enlarged collar, so (i) assigns them to the original sheet. Conversely every such branch stabilizer preserves the sheet. We have proved generation by these whole stabilizers, not merely an assertion about individual elements of them.
There are finitely many branch images in the controlled neighborhood: E E E f f f Z Z Z
Figure 1 summarizes the distinction between vertex chambers and positive-dimensional face chambers in Lemma 3.5.
Figure 1. Schematic coarse neighborhoods for selective cubulation. The closed mid-halfspace chamber E v E_v E v E σ E_\sigma E σ E v E_v E v p − 1 ( v ) p^{-1}(v) p − 1 ( v )
Construct Y ~ \widetilde{Y} Y Y Y Y E σ E_\sigma E σ
q : Y ~ ⟶ Y (6) q:\widetilde{Y}\longrightarrow Y \tag*{(6)} q : Y ⟶ Y (6) and an action of P P P H ‾ \overline{H} H
Lemma 3.6 (Cubical branched model). The complex Y ~ \widetilde{Y} Y P P P q q q Y Y Y
Proof. First consider links. Two adjacent lifted edge germs project to distinct germs, since they occur on a cube projecting isomorphically to a base cube. Hence a clique projects to a clique with distinct germs. An individual lifted edge determines a unique lift of every positive-dimensional containing cube by the positive-face bijections of Lemma 3.5. The base flag condition and these bijections therefore place all germs of the clique on its unique lifted cube. Links are simplicial and flag.
We now establish simple connectivity; local flagness alone would not suffice. There is a useful canonical-lifting rule. Suppose b b b X ~ \widetilde{X} X a a a C C C
d ( a , f π b ) ≤ 2 r + 10 r 0 . (7) d(a,f\pi b)\leq2r+10r_0. \tag*{(7)} d ( a , f π b ) ≤ 2 r + 10 r 0 . (7) By the choice of D D D p a pa p a σ \sigma σ b b b E σ E_\sigma E σ σ \sigma σ p a pa p a
This lift is continuous. Near a parameter mapping into an open face, local finiteness downstairs confines nearby values to cofaces of that face. Local path connectivity of the parameter space and the collar assignment in Lemma 3.5 make their sheet restrictions agree on the face. Although there can be infinitely many coface sheets upstairs, their points approach this same face lift: distance to the face is bounded by the corresponding coordinate distance inside a unit cube, uniformly in dimension. Thus the rule is continuous in the cubical path metric. In particular it applies to a = f π b a=f\pi b a = f π b
Conversely, let α \alpha α Y ~ \widetilde{Y} Y c e ∈ E e c_e\in E_e c e ∈ E e z e ∈ X ~ z_e\in\widetilde{X} z e ∈ X r r r E v E_v E v b b b X ~ \widetilde{X} X
Project the image f π b f\pi b f π b E v E_v E v c e c_e c e E v E_v E v v v v α \alpha α
Since X ~ \widetilde{X} X b b b a = f π b a=f\pi b a = f π b Y ~ \widetilde{Y} Y α \alpha α E v E_v E v Y ~ \widetilde{Y} Y [58 ] .
The action is cocompact because Y / H ˉ Y/\bar{H} Y / H ˉ B B B q q q
Completion of Proposition 3.3. Take the finitely many hyperplane orbit families of the compact special cover of C C C Y Y Y Y ~ \widetilde{Y} Y
Cut successively along the lifts of each family, within the convex components left by previous cuts. Disjoint two-sided tracks in a simply connected component give a tree splitting. The resulting graph-of-groups quotients are finite. Indeed, subdivide along the midplanes. A chamber and its incident side pieces are specified by cube-side flags up to the group action. At a cell only the hyperplanes cutting that cell can vary the choices of sides, so the number of choices is bounded in terms of the dimension. Branching at original vertices introduces no additional midplane choices.
A cutting trace is a convex part of a lifted hyperplane. It contains a dual-edge midpoint in its chamber: in a cube meeting the trace, fix the dual coordinate at its middle and choose the remaining corner coordinates towards the chamber’s defining sides. Thus the trace determines its containing hyperplane. A deck transformation preserving the trace preserves this hyperplane, acts trivially on it by Lemma 3.6, and is trivial because positive-cell sheet actions are free. Hence the trace stabilizer injects into H ˉ \bar{H} H ˉ
The trace projects to a convex subcomplex of the corresponding base hyperplane. Each earlier cutting hyperplane meeting the lifted hyperplane has a unique corresponding base trace; thus its side condition projects to the same convex halfspace condition downstairs. The stabilizer acts cocompactly on the trace by the finite flag description. Its image is consequently undistorted in Q ∗ Q_* Q ∗ P → H ˉ P \to\bar{H} P → H ˉ P P P P P P
After all families are cut, each component has one original cubical vertex. Its stabilizer projects trivially to H ˉ \bar{H} H ˉ H ˉ \bar{H} H ˉ Y Y Y
B = ∗ a retained R a . B = \mathop{*}_{a\ \text{retained}} R_a. B = ∗ a retained R a . Each accessible whole factor is one of its Kurosh factors, up to subgroup conjugacy. Killing these factors kills the entire sheet stabilizer by the proved generation statement. Hence there can be no remaining free summand or partial-factor summand. The vertex stabilizer is a finite free product of retained factors and is hyperbolic and virtually compact special.
The hierarchy has finitely many levels and finite quotients. Working upwards from its terminal groups shows that all vertex groups are finitely generated. The globally quasiconvex edge groups give quasiconvex vertex groups successively: replace excursions of an ambient path outside a vertex space by paths in the attaching edge cosets, using their ambient undistortion. The quasiconvex hierarchy theorem [66 ] therefore makes P P P H H H
Novikov realization along quotient chains
The geometric construction now supplies quotients that retain any prescribed finite set of filling-kernel factors. Our algebraic target is to realize any finite list of rational expressions, after finite-index restriction, in the ordinary Novikov rings for both signs of one real character. Applied to the entries of a division-ring chain contraction, this will give the two acyclicity statements needed for fibering. We first descend through an RFRS chain of the quotient until the remaining coefficients lie in D ( K ) D(K) D ( K )
For a real character χ : J → R \chi:J\to\mathbb{R} χ : J → R
Q ^ χ J = { ∑ g ∈ J a g g : # { g : a g ≠ 0 , χ ( g ) < c } < ∞ for every c ∈ R } . (8) \widehat{\mathbb{Q}}^{J}_{\chi}=\left\{\sum_{g\in J}a_{g}g:\#\{g:a_{g}\ne0,\ \chi(g)<c\}<\infty\text{ for every }c\in\mathbb{R}\right\}. \tag*{(8)} Q χ J = ⎩ ⎨ ⎧ g ∈ J ∑ a g g : # { g : a g = 0 , χ ( g ) < c } < ∞ for every c ∈ R ⎭ ⎬ ⎫ . (8) Our completion convention is towards positive height. The subgroup-field hypotheses embed D ( J ) D(J) D ( J ) D ( ker χ ) D(\ker\chi) D ( ker χ ) (1) embeds the rational field there; a nonzero series has a leading term, and its positive remainder has a positive minimum valuation, so geometric inversion is defined. This remains valid when χ ( J ) \chi(J) χ ( J ) R \mathbb{R} R
Definition 3.7. Let K ⊲ J K\lhd J K ⊲ J J / K J/K J / K ( J , K ) (J,K) ( J , K ) selective realization property if, for every finite-index preimage J ′ ≤ J J'\leq J J ′ ≤ J T ⊂ K T\subset K T ⊂ K H ≤ J ′ H\leq J' H ≤ J ′ J ′ J' J ′ U ≤ K U\leq K U ≤ K T T T H H H U U U
Proposition 3.3 supplies this property in Theorem 3.1: a finite set in K K K (3) , and its selective quotient is virtually compact special, hence virtually RFRS [1 ] . More explicitly, let ρ : H → L \rho:H\to L ρ : H → L S S S H \ J ′ H\backslash J' H \ J ′ H H H
h ⟼ ( ρ ( s − 1 h s ) ) s ∈ S ( h ∈ H ) h\longmapsto\left(\rho\left(s^{-1}hs\right)\right)_{s\in S}\qquad(h\in H) h ⟼ ( ρ ( s − 1 h s ) ) s ∈ S ( h ∈ H ) a homomorphism on the common domain H H H s s s s U s − 1 sUs^{-1} s U s − 1
Lemma 3.8 (Controlled linearization). A rational expression over a coefficient ring in a division ring has a representation p A − 1 q pA^{-1}q p A − 1 q A A A p , q p,q p , q A − 1 A^{-1} A − 1
Proof. A polynomial f f f ( 1 − f 0 0 1 ) \begin{pmatrix}1-f&0\\0&1\end{pmatrix} ( 1 − f 0 0 1 ) E = p A − 1 q E=pA^{-1}q E = p A − 1 q F = r B − 1 s F=rB^{-1}s F = r B − 1 s
( A − q r 0 B ) − 1 = ( A − 1 A − 1 q r B − 1 0 B − 1 ) \begin{pmatrix} A&-qr\\0&B\end{pmatrix}^{-1}
=
\begin{pmatrix} A^{-1}&A^{-1}qrB^{-1}\\0&B^{-1}\end{pmatrix} ( A 0 − q r B ) − 1 = ( A − 1 0 A − 1 q r B − 1 B − 1 ) for the product. For E − 1 E^{-1} E − 1 A A A
M = ( 0 p q A ) , M − 1 = ( − E − 1 E − 1 p A − 1 A − 1 q E − 1 A − 1 − A − 1 q E − 1 p A − 1 ) . M=\begin{pmatrix}0 & p \\ q & A\end{pmatrix},\qquad
M^{-1}=\begin{pmatrix}-E^{-1} & E^{-1}pA^{-1} \\ A^{-1}qE^{-1} & A^{-1}-A^{-1}qE^{-1}pA^{-1}\end{pmatrix}. M = ( 0 q p A ) , M − 1 = ( − E − 1 A − 1 q E − 1 E − 1 p A − 1 A − 1 − A − 1 q E − 1 p A − 1 ) . The displayed formulas prove the assertion inductively. In (3.13) the new inverse is exactly the parent inverse already present in the original expression.
The data at a descent stage are finitely many square polynomial matrices invertible over the current division ring, together with chosen rational expressions for all entries of their inverses. Linearization puts a finite list of scalar expressions in this form. Truncating a character expansion records finitely many coefficient entries. Solving those coefficient problems means realizing their chosen rational values in both ordinary Novikov rings for one later character. The next lemma proves that this reduction terminates; the following lemma explains how to recover the original inverses.
Lemma 3.9 (Finite descent of inverse depth). Let J / K J/K J / K
J = H 0 ≥ H 1 ≥ ⋯ , ⋂ i H i = K . J=H_0\ge H_1\ge\cdots,\qquad\bigcap_i H_i=K. J = H 0 ≥ H 1 ≥ ⋯ , i ⋂ H i = K . Let N i ◃ H 0 N_i\triangleleft H_0 N i ◃ H 0 H i / K H_i/K H i / K N i ≤ H i + 1 N_i\le H_{i+1} N i ≤ H i + 1 N i N_i N i H i + 1 H_{i+1} H i + 1 D ( K ) D(K) D ( K )
Proof. Inverse depth is measured relative to D ( K ) D(K) D ( K )
For one character, write an operand with leading monomial L L L L ( 1 + T ) L(1+T) L ( 1 + T ) T T T ∑ n ≥ 0 ( − T ) n L − 1 \sum_{n\ge0}(-T)^nL^{-1} ∑ n ≥ 0 ( − T ) n L − 1
There is a more precise invariant. If a coefficient retains the old maximum depth d > 0 d>0 d > 0 d d d d d d d − 1 d-1 d − 1 d d d
Choose one right normalizing monomial for each slice. Replacing support elements g , h g,h g , h g u − 1 , h u − 1 gu^{-1},hu^{-1} g u − 1 , h u − 1 g h − 1 gh^{-1} g h − 1 d d d K K K
Restriction to the next subgroup does not mix the coefficients: for c ∈ D ( N i ) c\in D(N_i) c ∈ D ( N i ) t t t H i + 1 \ H i H_{i+1}\backslash H_i H i + 1 \ H i
t c = ( t c t − 1 ) t , t c t − 1 ∈ D ( N i ) ⊂ D ( H i + 1 ) . tc=(tct^{-1})t,\qquad tct^{-1}\in D(N_i)\subset D(H_{i+1}). t c = ( t c t − 1 ) t , t c t − 1 ∈ D ( N i ) ⊂ D ( H i + 1 ) . Thus the coefficient operator is diagonal by conjugation on these coset coordinates. Controlled linearization also preserves the invariant by Lemma 3.8.
The bottom support differences at stage i i i H i / K H_i/K H i / K D ( K ) D(K) D ( K )
The qualification about maximal depth is essential to this argument. New denominator supports can appear after the depth drops; the construction expressly starts a fresh finite list at that point. It never asserts that one support list controls all later depths.
A given RFRS chain may be replaced by one normal in its initial group. Indeed, intersect its finitely many conjugates at each stage; the rational abelianization kernel of a subgroup maps into that kernel in every containing conjugate. The RFRS containment therefore persists under these cores. In the finitely generated setting of Proposition 3.11, characters with kernel exactly N i N_i N i H i / N i H_i/N_i H i / N i R \mathbb{R} R D ( K ) D(K) D ( K ) Q K \mathbb{Q}K Q K
Lemma 3.10 (Recombination by perturbation). Suppose a finite stage of the preceding procedure used a character χ i \chi_i χ i H i H_i H i A A A B + B_+ B + B − B_- B − A − 1 A^{-1} A − 1 I − A B ± I-AB_\pm I − A B ± I − B ± A I-B_\pm A I − B ± A ± χ i \pm\chi_i ± χ i A A A B + B_+ B + B − B_- B − F F F ψ \psi ψ F F F
χ i ∣ F + ϵ ψ \chi_i|_F+\epsilon\psi χ i ∣ F + ϵ ψ for every sufficiently small positive ϵ \epsilon ϵ
Proof. Let A A A B = A − 1 B=A^{-1} B = A − 1 B + B_+ B + B − B_- B − D ( N i ) D(N_i) D ( N i )
I − A B ± , I − B ± A I-AB_\pm,\qquad I-B_\pm A I − A B ± , I − B ± A have strictly positive weights in their respective directions.
First we check the support needed to compare the two characters. This is necessary even when a later finite-index cut does not contain N i N_i N i t j t_j t j F \ H i F\backslash H_i F \ H i t j t_j t j t k t_k t k u u u
u + χ i ( t j ) − χ i ( t k ) . u+\chi_i(t_j)-\chi_i(t_k). u + χ i ( t j ) − χ i ( t k ) . Its coefficient lies in one coset of D ( F ∩ N i ) D(F\cap N_i) D ( F ∩ N i ) U = F ∩ N i U=F\cap N_i U = F ∩ N i U U U ψ \psi ψ ψ \psi ψ U U U
On F ∩ N i F \cap N_i F ∩ N i (3.15) is ϵ ψ \epsilon\psi ϵ ψ (3.17) when measuring an error entry. There are finitely many positive old-height slices, and their ± ψ \pm\psi ± ψ ϵ \epsilon ϵ
Offsets telescope in matrix products. Hence powers of either error in (3.16) have valuations tending to positive infinity, even in the unadjusted coordinates. Geometric inversion provides both a left and a right inverse for A A A ϵ \epsilon ϵ
Proposition 3.11 (Progressive characters). Let J J J K ◃ J K \triangleleft J K ◃ J Q J \mathbb{Q}J Q J F F F F F F
In particular, if a finite complex of finitely generated free Q J \mathbb{Q}J Q J D ( J ) D(J) D ( J )
Proof. Take the preimage H 0 H_0 H 0 J / K J/K J / K
Pull back a normal RFRS chain from H 0 / K H_0/K H 0 / K i i i (1) to expand each inverse matrix for both signs of a character χ i \chi_i χ i N i N_i N i (3.16) . Collect the finitely many coefficients of the matrix and its truncations; they lie in D ( N i ) D(N_i) D ( N i ) (3.14) , and linearize these scalar problems. Lemma 3.9 terminates this procedure with finitely many unresolved coefficients in D ( K ) D(K) D ( K )
Choose a finite set of elements of K K K H H H H p H_p H p
This cut can mix coefficient entries, so we describe it explicitly. For a scalar c ∈ D ( U 0 ) c \in D(U_0) c ∈ D ( U 0 ) H p H_p H p U 0 U_0 U 0 g j g_j g j u j ∈ U 0 u_j \in U_0 u j ∈ U 0 c c c u j c u_jc u j c
D ( U 0 ∩ g − 1 H p g ) D(U_0 \cap g^{-1}H_pg) D ( U 0 ∩ g − 1 H p g ) and then conjugating by g g g
Start the depth argument afresh with these new expressions. Use an RFRS chain in the new quotient and ordinary rational group polynomials. Work separately inside each support subgroup, choosing its own monomial transporters when taking a height coefficient. Later coefficients remain in conjugates of that support subgroup. Once a coefficient lies in the current character kernel, restriction to the next stage only conjugates it, because that whole kernel is contained in the next subgroup. Thus the maximal-depth ancestry argument applies exactly as before. The new quotient map is injective on each support subgroup, so its normal RFRS chain separates the finite nonidentity support lists. This time the terminal problems are ordinary group-ring polynomials, with no unresolved division-field coefficients. They are realized for the zero character if necessary.
Now reconstruct the inverse problems backwards, through both finite chains and the selective switch, using Lemma 3.10. At each step use both signs’ residual lists and choose one sufficiently small positive perturbation. The result is a single character on a common finite-index subgroup and compatible realizations of all original expressions for both of its signs.
For the chain-complex assertion, include the entries of a D ( J ) D(J) D ( J ) Q F \mathbb{Q}F Q F
Corollary 3.12 (Open Novikov regions for RFRS groups). Let J J J Q J \mathbb{Q}J Q J D ( J ) D(J) D ( J ) F F F Hom ( F , R ) \operatorname{Hom}(F,\mathbb{R}) Hom ( F , R )
Proof. We use the descent above with K = 1 K=1 K = 1 J J J [36 ] , Theorem 5.2. To retain the present group generality, we first justify the ambient characters and the character-field expansion without finite generation.
Choose a finite-index RFRS subgroup H 0 H_0 H 0 ( H i ) (H_i) ( H i ) H 0 ↪ ∏ i H 0 / H i H_0 \hookrightarrow\prod_i H_0/H_i H 0 ↪ ∏ i H 0 / H i ∣ H 0 ∣ ≤ ∣ R ∣ \lvert H_0\rvert\leq\lvert\mathbb{R}\rvert ∣ H 0 ∣ ≤ ∣ R ∣ N i N_i N i H i H_i H i H i / N i H_i/N_i H i / N i dim Q R \dim_{\mathbb{Q}}\mathbb{R} dim Q R R \mathbb{R} R H i H_i H i N i N_i N i N i ≤ H i + 1 N_i \leq H_{i+1} N i ≤ H i + 1
For any real character χ \chi χ H H H N = ker χ N=\ker\chi N = ker χ D ( N ) ∗ ( H / N ) D(N) * (H/N) D ( N ) ∗ ( H / N ) D ( H ) D(H) D ( H ) H / N H/N H / N D ( H ) D(H) D ( H ) D ( N ) D(N) D ( N ) Q H \mathbb{Q}H Q H D ( H ) D(H) D ( H ) D ( H ) D(H) D ( H ) (1) in this generality.
The same skew-series realization applies: a nonempty support finite below every real threshold has a least term, and each strictly positive remainder has an individual positive minimum valuation. These facts justify geometric inversion even when the character image is dense or not finitely generated.
Start with the finitely many entries of division-ring chain contractions. The descent in Lemma 3.9 uses only finite rational expressions, finite retained coefficient lists and finite-index transversals. Normal residuality removes its finite support-difference lists, so it terminates with coefficients over D ( 1 ) = Q D(1)=\mathbb{Q} D ( 1 ) = Q
For openness, linearize the finitely many contraction entries over the ordinary group ring and truncate their realized inverse matrices to finite group-ring matrices. The left and right inverse errors have finitely supported strictly positive valuations. Their finitely many strict evaluation inequalities define an open neighborhood in the pointwise topology, and geometric inversion supplies the contractions throughout it. Intersect the neighborhood for the positive sign with the negative of the neighborhood for the negative sign, and then take the union with its negative. This is the required nonempty antipodally invariant open set.
On restriction to any deeper finite-index subgroup, a Novikov series remains summable in its finite coset coordinates, with only fixed monomial offsets. Repeat the finite truncation argument there to obtain openness in the larger character space.
From two-sided Novikov acyclicity to a free kernel
Lemma 3.13 (The degree-one Novikov implication). Let J J J χ : J → R \chi:J\to\mathbb{R} χ : J → R Q J χ \mathbb{Q}J^\chi Q J χ χ \chi χ
Proof. We include the rational-coefficient argument in the positive-completion convention of [36 ] , Theorem 3.11; Sikorav established the integral-coefficient criterion [60 ] . Choose a symmetric generating set containing a generator t t t χ ( t ) > 0 \chi(t)>0 χ ( t ) > 0
c = ∑ j ≥ 0 [ t j , t j + 1 ] c=\sum_{j\geq0}[t^j,t^{j+1}] c = j ≥ 0 ∑ [ t j , t j + 1 ] has boundary − [ 1 ] -[1] − [ 1 ] x x x
[ 1 , x ] + x c − c [1,x]+xc-c [ 1 , x ] + x c − c is a cycle, hence bounds a completed rational two-chain in the finite presentation complex. Truncate that two-chain sufficiently high. Since relators have bounded height offsets, its boundary cancels every low term of the displayed cycle. Truncate both rays at a common sufficiently large exponent k k k
[ x t k ] − [ t k ] . [xt^k]-[t^k]. [ x t k ] − [ t k ] . The component of its edge support containing one endpoint must contain the other: the total boundary coefficient in a finite connected component is zero. Thus t k t^k t k x t k xt^k x t k x x x
Translate these finitely many paths to replace each edge g → g x g\to gx g → g x g t k gt^k g t k g x t k gxt^k g x t k t t t
The finite paths and their strict height inequalities persist in a neighborhood of χ \chi χ − χ -\chi − χ [9 ] , Theorem B1, also stated in [36 ] , Theorem 3.10, says that the kernel of its integral representative is finitely generated.
Lemma 3.14 (A finitely presented kernel in dimension two). Suppose cd J ≤ 2 \operatorname{cd} J \le2 cd J ≤ 2 and J J J maps onto Z \mathbb{Z} Z with finitely presented kernel M M M Then M M M is free.
Proof. We have cd M ≤ 2 \operatorname{cd} M \le2 cd M ≤ 2 cd M = 2 \operatorname{cd} M = 2 cd M = 2 Z \mathbb{Z} Z Z M \mathbb{Z}M Z M
H 2 ( M ; Z M ) ≠ 0. H^2(M;\mathbb{Z}M) \ne0. H 2 ( M ; Z M ) = 0. As an M M M Z J \mathbb{Z}J Z J Z M \mathbb{Z}M Z M J / M = Z J/M = \mathbb{Z} J / M = Z H 2 ( M ; Z J ) H^2(M;\mathbb{Z}J) H 2 ( M ; Z J ) J / M J/M J / M
H 3 ( J ; Z J ) ≅ H 1 ( Z ; H 2 ( M ; Z J ) ) ≠ 0 , H^3(J;\mathbb{Z}J) \cong H^1(\mathbb{Z};H^2(M;\mathbb{Z}J)) \ne0, H 3 ( J ; Z J ) ≅ H 1 ( Z ; H 2 ( M ; Z J )) = 0 , contrary to cd J ≤ 2 \operatorname{cd} J \le2 cd J ≤ 2 [14 , 8 ] . It follows that cd M ≤ 1 \operatorname{cd} M \le1 cd M ≤ 1 [61 , 64 ] makes M M M
Proof of Theorem 3.1. First suppose G ∗ G_* G ∗ G ∗ G_* G ∗ G ∗ G_* G ∗ F F F
In the remaining case choose the cofinite good filling (3.2). Proposition 3.3 gives the selective realization property, and Proposition 3.11 applied to the same chains again supplies two-sided Novikov contractions on a finite-index subgroup F F F
In either case the restricted group-ring chain complex has H 0 ≅ Q H_0 \cong\mathbb{Q} H 0 ≅ Q F F F Q F \mathbb{Q}F Q F H 0 H_0 H 0 F F F F → Z F \to\mathbb{Z} F → Z
Coherence makes this kernel finitely presented. Lemma 3.14 makes it free, and finite generation makes its rank finite. The extension splits by choosing a lift of a generator of Z \mathbb{Z} Z [27 ] ; the elementary rank-zero case is immediate. Agol’s theorem [2 ] now gives virtual compact specialness, which passes to the finite-index overgroup G ∗ G_* G ∗
Hyperbolic ascending tori
For an injective endomorphism ϕ : F → F \phi:F \to F ϕ : F → F
J ϕ = ⟨ F , t ∣ t − 1 g t = ϕ ( g ) ( g ∈ F ) ⟩ . J_\phi= \langle F,t \mid t^{-1}gt = \phi(g)\ (g \in F)\rangle. J ϕ = ⟨ F , t ∣ t − 1 g t = ϕ ( g ) ( g ∈ F )⟩ . Its height homomorphism sends t t t 1 1 1 F F F free splitting tree means a minimal simplicial tree with trivial edge stabilizers and finite quotient; its vertex stabilizers are free factors. A path map of trees is alignment preserving if it maps each segment monotonically onto a segment, allowing constant subsegments.
Theorem 4.1. Let F F F ϕ : F → F \phi:F\to F ϕ : F → F J ϕ J_{\phi} J ϕ J ϕ J_{\phi} J ϕ
We prove the theorem by induction on rk F \operatorname{rk} F rk F BS ( 1 , n ) \operatorname{BS}(1,n) BS ( 1 , n ) n ≠ 0 n \ne0 n = 0 ϕ \phi ϕ [27 ] and Agol’s theorem [2 ] . For a nonsurjective injection, we either reduce rank or construct a noncontracting map of a free splitting tree. In the expanding case, its associated real-tree action gives a finite family of point stabilizers with nontrivial height-zero part. They are virtually ascending tori of smaller rank and will serve as peripherals. Filling their height-zero parts makes the base virtually free; a finite graph cover then separates the expanding map into irreducible train-track blocks. We remove any failure of fundamental-group injectivity in those blocks before applying Hagen–Wise. This supplies the good fillings for sec:3.1 . The nonexpanding case is handled directly by a quasiconvex hierarchy.
The ascending result is also an input to the filling construction in Section 10. After proving it, we recover its stronger virtual finite-rank free-by-Z \mathbb{Z} Z
Finiteness hypotheses and elementary changes of model
Lemma 4.2. An ascending torus of an injective finite-rank free-group endomorphism is torsion-free, locally indicable, coherent, residually finite, and has a finite aspherical complex of dimension at most two. It has the Hughes-free subgroup-field setup used in Theorem 3.1, and its finite cellular chain complex becomes acyclic over its rational Linnell field.
Proof. The height kernel is an increasing union of conjugates of F F F Z \mathbb{Z} Z [22 ] . Residual finiteness is due to Borisov–Sapir [10 ] .
Represent ϕ \phi ϕ
The height kernel belongs to Linnell’s class generated by free groups under directed unions; its extension by Z \mathbb{Z} Z [38 , 32 ] ; the locally indicable affiliated realization and compatible rational subfields are given by [35 ] and [33 ] . In particular the field embeds in the height skew-series division ring with coefficients in the kernel field, with either order on height.
For completeness, take a basis x 1 , … , x r x_1,\ldots,x_r x 1 , … , x r F F F t − 1 x i t ϕ ( x i ) − 1 t^{-1}x_it\phi(x_i)^{-1} t − 1 x i tϕ ( x i ) − 1
M = t − 1 I − ( ∂ ϕ ( x i ) ∂ x j ) i , j . M=t^{-1}I-\left(\frac{\partial\phi(x_i)}{\partial x_j}\right)_{i,j}. M = t − 1 I − ( ∂ x j ∂ ϕ ( x i ) ) i , j . After removing the invertible initial factor t − 1 t^{-1} t − 1 I I I M M M M M M t − 1 t-1 t − 1 1 − ( r + 1 ) + r = 0 1-(r+1)+r=0 1 − ( r + 1 ) + r = 0
We will use the following elementary observation twice. It applies even when a graph map is not injective on fundamental groups.
Lemma 4.3. Suppose based maps f : X → X f:X\to X f : X → X and g : Y → Y g:Y\to Y g : Y → Y of connected finite graphs admit based maps q : X → Y q:X\to Y q : X → Y and i : Y → X i:Y\to X i : Y → X such that
q f = g q , i g = f i , i q = f m , q i = g m qf=gq,\qquad ig=fi,\qquad iq=f^{m},\qquad qi=g^{m} q f = g q , i g = f i , i q = f m , q i = g m for some m ≥ 1 m\ge1 m ≥ 1 Their mapping-torus groups are isomorphic, by an isomorphism preserving height.
Proof. The commuting maps induce homomorphisms of the mapping-torus presentations, sending stable letter to stable letter. Their composites are induced by f m f^{m} f m g m g^{m} g m
An expanding relative tree model
Expanding relative immersions for injective, nonsurjective free-group endomorphisms were constructed by Mutanguha [51 ] . The following form is tailored to the induction on rank.
Lemma 4.4. Let ϕ : F → F \phi:F\to F ϕ : F → F be injective and nonsurjective, with rk F ≥ 2 \operatorname{rk} F\ge2 rk F ≥ 2 After replacing F F F by an iterated image, one of the following holds.
(i) A finite-index height subgroup of J ϕ J_\phi J ϕ is an ascending torus of an injection of a free group of rank strictly smaller than rk F \operatorname{rk} F rk F
(ii) There is a nontrivial free splitting tree T T T and a ϕ \phi ϕ -equivariant embedding f : T → T f:T\to T f : T → T that is a homothety of scale λ ≥ 1 \lambda\ge1 λ ≥ 1 The edge-transition matrix is irreducible, every vertex stabilizer is a proper free factor, and the quotient graph is finite.
Proof. We compare successive iterated-image core graphs using the lengths of their arcs between branch vertices. Collapsing the arcs whose lengths stay bounded gives two trees and two maps: substitution identifies the collapsed trees, while their original inclusion gives a possibly collapsing map back. Their composition yields the required self-map. Its transition diagram will either lose an edge orbit, allowing rank reduction, or supply a recurrent irreducible part.
Fix the unit-edge Cayley tree S S S F F F F n = ϕ n ( F ) F_n=\phi^n(F) F n = ϕ n ( F ) S n ⊂ S S_n\subset S S n ⊂ S S n S_n S n natural arcs . The finite core graph F n \ S n F_n\backslash S_n F n \ S n r = rk F r=\operatorname{rk} F r = rk F 3 r − 3 3r-3 3 r − 3
Substitution by the fixed injection ϕ \phi ϕ S S S F F F S n S_n S n S n + 1 S_{n+1} S n + 1 ϕ \phi ϕ
S n ⟶ S n + 1 S_n\longrightarrow S_{n+1} S n ⟶ S n + 1 with constants independent of n n n
Choose a subsequence of pairs
U k = S n k , V k = S n k + 1 U_k = S_{n_k}, \qquad V_k = S_{n_k+1} U k = S n k , V k = S n k + 1 on which the two finite reduced core-graph types are fixed and each of their natural-edge lengths is either bounded or tends to infinity. There is one bound B B B ℓ k \ell_k ℓ k U k U_k U k V k V_k V k
For sufficiently large k k k ϕ \phi ϕ ℓ k \ell_k ℓ k [ v , w ] [v,w] [ v , w ] v ′ , w ′ v',w' v ′ , w ′ z z z [ v ′ , w ′ ] [v',w'] [ v ′ , w ′ ] [ v , w ] [v,w] [ v , w ] v v v w w w z z z v ′ v' v ′ w ′ w' w ′ [ v ′ , w ′ ] [v',w'] [ v ′ , w ′ ] ℓ k \ell_k ℓ k
The collapsed tree is nontrivial for some such subsequence. Otherwise all natural arcs have uniformly bounded lengths along an infinite subsequence. There would then be only finitely many unbased labelled core graphs, so two nested groups F n , F n + j F_n,F_{n+j} F n , F n + j g A g − 1 ⊆ A gAg^{-1}\subseteq A g A g − 1 ⊆ A g − i S A g^{-i}S_A g − i S A S A S_A S A S A S_A S A A A A S A S_A S A g − i A g^{-i}A g − i A g j ∈ A g^j\in A g j ∈ A j > 0 j>0 j > 0 F n = F n + j F_n=F_{n+j} F n = F n + j F = ϕ j ( F ) F=\phi^j(F) F = ϕ j ( F )
Inclusion V k ⊂ U k V_k\subset U_k V k ⊂ U k V k V_k V k U k U_k U k B < ℓ k B<\ell_k B < ℓ k k k k n = n k n=n_k n = n k T U , T V T_U,T_V T U , T V F n F_n F n F n + 1 F_{n+1} F n + 1
T U → Φ k T V → I k T U . T_U \xrightarrow{\Phi_k} T_V \xrightarrow{I_k} T_U. T U Φ k T V I k T U . Here Φ k \Phi_k Φ k ϕ : F n ⟶ ∼ F n + 1 \phi:F_n\overset{\sim}{\longrightarrow}F_{n+1} ϕ : F n ⟶ ∼ F n + 1 I k I_k I k F n + 1 ≤ F n F_{n+1}\leq F_n F n + 1 ≤ F n I k Φ k I_k\Phi_k I k Φ k ϕ \phi ϕ F n F_n F n T U T_U T U k k k F n F_n F n F F F F n F_n F n F F F
If an iterate of the whole tree misses an edge orbit, its connected image lies in one component of the complement of that orbit. The image group ϕ m ( F ) \phi^m(F) ϕ m ( F ) C C C
ϕ m ( F ) ≤ C , ϕ m ( C ) ≤ C . \phi^m(F) \le C,\qquad\phi^m(C) \le C. ϕ m ( F ) ≤ C , ϕ m ( C ) ≤ C . The ascending torus of ϕ m ∣ C \phi^m|_C ϕ m ∣ C m m m
Otherwise choose a source strongly connected component in the finite edge-transition diagram. If it were nonrecurrent, it would consist of one vertex with no self-loop and no incoming arrow; that edge orbit would already be missing from the image of the whole tree. The chosen component is therefore recurrent. No incoming arrows means that each edge outside it maps entirely through edges outside it. Their union is an invariant forest in the tree, so collapsing it gives a well-defined self-map. Each remaining edge has a transition inside the recurrent component and is not collapsed. Alignment preservation also survives: lift a segment through the collapsed vertex components, apply alignment preservation before collapse, and delete the collapsed subsegments. No reduced path can acquire backtracking in this operation. The transition matrix is now irreducible. Its Perron eigenvector assigns positive lengths to the finitely many edge types, and linear parametrization makes the map multiply all path lengths by the Perron eigenvalue λ ≥ 1 \lambda\ge1 λ ≥ 1
Point stabilizers and their quasiconvexity
Assume alternative (ii) of Lemma 4.4. Over each vertex b b b B \mathcal{B} B T b T_b T b T T T B \mathcal{B} B f f f T \mathcal{T} T λ \lambda λ
Lemma 4.5. The height-zero subgroup of J ϕ J_\phi J ϕ T \mathcal{T} T J ϕ J_\phi J ϕ x x x A x A_x A x F F F λ > 1 \lambda> 1 λ > 1
Proof. For a height-zero element and a compact arc, choose a sufficiently high ancestor copy containing both endpoints and whose base group contains the element. A nontrivial element fixing that arc would fix an edge of the free splitting, which is impossible.
Suppose a nontrivial base element fixes x x x T b T_b T b T \mathcal{T} T x x x x x x T b T_b T b x ∈ T b x \in T_b x ∈ T b
Let C b , x C_{b,x} C b , x x x x b b b b b b C b , x ≠ 1 C_{b,x} \ne1 C b , x = 1 B \mathcal{B} B ( b , x ) (b,x) ( b , x ) J ϕ J_\phi J ϕ s ∈ A x s \in A_x s ∈ A x b b b A x A_x A x x x x ker ( h ∣ A x ) \ker(h|_{A_x}) ker ( h ∣ A x ) C b , x C_{b,x} C b , x h ( s ) Z h(s)\mathbb{Z} h ( s ) Z C b , x C_{b,x} C b , x s s s A x A_x A x C b , x C_{b,x} C b , x
Finally suppose λ > 1 \lambda> 1 λ > 1
If λ = 1 \lambda= 1 λ = 1 [66 ] proves Theorem 4.1 in this case.
We henceforth assume λ > 1 \lambda> 1 λ > 1
Lemma 4.6. Each subgroup A x A_x A x J ϕ J_\phi J ϕ rk F \operatorname{rk} F rk F
Proof. Realize the free splitting as a finite graph with a bouquet at each vertex carrying its vertex free factor; its other edges realize the edges of the splitting. Represent f f f
Use this representative to form the ordinary tree of graph spaces for the torus. Its one-skeleton X X X C b , x ≠ 1 C_{b,x} \ne1 C b , x = 1 W b W_b W b x x x Y x Y_x Y x A x A_x A x Y x Y_x Y x
Project each retained horizontal row to its convex subtree W b W_b W b Y x Y_x Y x
It remains to treat a component of omitted rows. It attaches to the retained subtree of B \mathcal{B} B d → b d \to b d → b C d , x = 1 C_{d,x} = 1 C d , x = 1 W b W_b W b x x x x x x x x x C d , x ≠ 1 C_{d,x} \ne1 C d , x = 1 W b W_b W b
Map the entire omitted component to a point within bounded distance of these gates. The resulting map on the vertices of X X X Y x Y_x Y x Y x Y_x Y x A x A_x A x A x A_x A x J ϕ J_\phi J ϕ
A finite malnormal family of quasiconvex subgroups of a hyperbolic group is a relatively hyperbolic peripheral structure [11 ] . Each A x A_x A x
Deep fillings with train-track finite covers
Lemma 4.7. Assume the inductive hypothesis below rank rk F \operatorname{rk} F rk F
J ˉ = J ϕ / ⟨ ⟨ N x ⟩ ⟩ \bar{J} = J_\phi/\langle\langle N_x\rangle\rangle J ˉ = J ϕ / ⟨⟨ N x ⟩⟩ that is hyperbolic, has finite-by-cyclic peripheral images, and has a finite-index subgroup equal to the mapping-torus group of an expanding train-track map p : V → V p: V \to V p : V → V p p p
Proof. Choose a finite-index normal subgroup H ◃ J ϕ H \triangleleft J_\phi H ◃ J ϕ
N x = H ∩ ker h ∩ A x . N_x = H \cap\ker h \cap A_x. N x = H ∩ ker h ∩ A x . Residual finiteness permits H H H ker ( h ∣ A x ) \ker(h|_{A_x}) ker ( h ∣ A x ) A x / N x A_x/N_x A x / N x A x / N x A_x/N_x A x / N x [55 ] gives exact peripheral reduction and a relatively hyperbolic quotient with these peripheral images. As the peripheral images are hyperbolic, J ˉ \bar{J} J ˉ
Let C v C_v C v F ˉ \bar{F} F ˉ C v / ( C v ∩ H ) C_v/(C_v \cap H) C v / ( C v ∩ H ) F F F T f i n T^{\mathrm{fin}} T fin N x N_x N x J ˉ \bar{J} J ˉ F ˉ \bar{F} F ˉ T f i n T^{\mathrm{fin}} T fin
We impose one additional finite avoidance condition that makes every iterate of every edge reduced in T f i n T^{\mathrm{fin}} T fin f f f
For each seed turn whose directions ever enter the same oriented edge orbit, choose the first such derivative. Its two directions differ by a nonidentity multiplier in their vertex stabilizer. Add that multiplier to the finite avoidance set for H H H ϕ \phi ϕ ϕ \phi ϕ J ϕ J_\phi J ϕ H H H H H H
Put D = F / ( F ∩ H ) D=F/(F\cap H) D = F / ( F ∩ H ) σ \sigma σ F ˉ → D \bar{F}\to D F ˉ → D T f i n T^{\mathrm{fin}} T fin V V V p : V → V p:V\to V p : V → V
p ( d z ) = σ ( d ) p ( z ) ( d ∈ D ) . p(dz)=\sigma(d)p(z) \qquad(d\in D). p ( d z ) = σ ( d ) p ( z ) ( d ∈ D ) . All edges expand, since their relative lengths are multiplied by λ > 1 \lambda>1 λ > 1 F ˉ \bar{F} F ˉ σ \sigma σ J ˉ → D ⋊ σ Z \bar{J}\to D\rtimes_\sigma\mathbb{Z} J ˉ → D ⋊ σ Z ∣ D ∣ |D| ∣ D ∣ F ˉ \bar{F} F ˉ p ∗ p_* p ∗
Finally consider reachability of unoriented edges in V V V α \alpha α β \beta β β \beta β d α d\alpha d α d ∈ D d\in D d ∈ D α → d α \alpha\to d\alpha α → d α m m m
D ⟶ D , s ⟼ σ m ( s ) d . D\longrightarrow D,\qquad s\longmapsto\sigma^m(s)d. D ⟶ D , s ⟼ σ m ( s ) d . Its orbit of the identity closes. The resulting closed chain of occurrences passes through β \beta β β \beta β α \alpha α
Cubulating the filled groups
The next lemma treats the possible failure of fundamental-group injectivity in the preceding finite graph model.
Lemma 4.8. Let p : V 0 → V 0 p:V_0\to V_0 p : V 0 → V 0 p ∗ p_* p ∗
Proof. The free groups P m = p ∗ m ( π 1 V 0 ) P_m=p_*^m(\pi_1V_0) P m = p ∗ m ( π 1 V 0 ) m ≥ 1 m\ge1 m ≥ 1 p ∗ : P m → P m + 1 p_*:P_m\to P_{m+1} p ∗ : P m → P m + 1 P m P_m P m
Let ρ : V ^ → V 0 \rho:\widehat{V}\to V_0 ρ : V → V 0 P m P_m P m p m p^m p m q : V 0 → V ^ q:V_0\to\widehat{V} q : V 0 → V p p p p ^ : V ^ → V ^ \widehat{p}:\widehat{V}\to\widehat{V} p : V → V p ^ q = q p \widehat{p}q=qp p q = q p W = q ( V 0 ) W=q(V_0) W = q ( V 0 ) P m P_m P m q ∗ q_* q ∗ P m P_m P m p W = p ^ ∣ W p_W=\widehat{p}|_W p W = p ∣ W p p p
The maps q : V 0 → W q:V_0\to W q : V 0 → W ρ ∣ W : W → V 0 \rho|_W:W\to V_0 ρ ∣ W : W → V 0 p p p p W p_W p W p m p^m p m p W m p_W^m p W m
We check irreducibility of p W p_W p W e ′ , f ′ e',f' e ′ , f ′ W W W e , f e,f e , f V 0 V_0 V 0 q q q k k k p k p^k p k f f f q p k ( a ) qp^k(a) q p k ( a ) a a a V 0 V_0 V 0 q ( f ) q(f) q ( f ) B B B n ≥ k n\ge k n ≥ k q p n ( e ) qp^n(e) q p n ( e ) p n + m ( e ) p^{n+m}(e) p n + m ( e ) p W n ( e ′ ) p_W^n(e') p W n ( e ′ ) q p n ( e ) qp^n(e) q p n ( e ) 2 B 2B 2 B f ′ f' f ′
The map p W p_W p W [26 , 26 ] : it is a fundamental-group-injective train-track map of a finite graph, all edges expand, its transition matrix is irreducible, and its mapping-torus group is hyperbolic. That theorem gives a free cocompact cubulation. Agol’s theorem [2 ] gives virtual compact specialness.
Lemma 4.9. The fillings in Lemma 4.7 are virtually compact special.
Proof. Use the finite-index graph torus supplied there, and pass to the height power for which all its blocks are primitive and its periodic vertices are fixed. At every vertex of V V V V V V p p p
The eventual image of this graph map is connected and contains all blocks, since each block map is primitive. Its remaining intervals and central vertices are periodic. Restriction to the eventual image does not change the mapping-torus group: inclusion and a sufficiently high power of the map satisfy Lemma 4.3. Connector orbits now give annuli joining the point tori and the block tori. This is a finite graph-of-groups decomposition over infinite cyclic groups. Each edge injection is detected by height, so is injective.
The total group is hyperbolic, as a finite-index subgroup of J ‾ \overline{J} J [66 ] now gives virtual compact specialness of the total group, and therefore of its finite-index supergroup J ‾ \overline{J} J
Proof of Theorem 4.1. The base cases and the automorphism case were disposed of at the start. Apply Lemma 4.4. Its rank-reduction alternative is settled by induction and finite-index invariance. In the tree-model alternative, the case λ = 1 \lambda=1 λ = 1 λ > 1 \lambda>1 λ > 1 J ϕ J_{\phi} J ϕ
Corollary 4.10. Let F F F ϕ : F → F \phi:F\to F ϕ : F → F J ϕ J_{\phi} J ϕ H H H
1 ⟶ F d ⟶ H ⟶ Z ⟶ 1 1\longrightarrow F_d\longrightarrow H\longrightarrow\mathbb{Z}\longrightarrow1 1 ⟶ F d ⟶ H ⟶ Z ⟶ 1 for some finite d ≥ 0 d\geq0 d ≥ 0 F d F_d F d d d d H ≅ F d ⋊ α Z H\cong F_d\rtimes_{\alpha}\mathbb{Z} H ≅ F d ⋊ α Z α \alpha α F d F_d F d
Proof. Theorem 4.1 makes J ϕ J_{\phi} J ϕ H H H H 0 H_0 H 0 Q \mathbb{Q} Q Q H \mathbb{Q}H Q H H 0 H_0 H 0
Lemma 3.13 gives an epimorphism H → Z H\to\mathbb{Z} H → Z H H H 1 ∈ Z 1\in\mathbb{Z} 1 ∈ Z
The epimorphism in this corollary need not be the restriction of the canonical height homomorphism of the original ascending presentation.
Actual splittings and row homology
We now work with a torsion-free hyperbolic primitive extension group. The filling criterion of Section 3 and the ascending mapping-torus conclusion of Section 4 are available. An ascending mapping torus always allows an injective, possibly nonsurjective, endomorphism. Quasiconvexity of a subgroup of a row means quasiconvexity in that row unless the ambient group is specified. Our immediate goal is to control the stabilizers of paths in the actual splitting. The row homology will provide a common finite-dimensional space into which their first homology groups inject, and hence the bound used to construct the periodic peripheral groups. We need two outputs. The first is a simultaneous homology injection for path stabilizers, including after restriction to a subgroup. The second is a quasiconvexity criterion for finitely generated periodic subgroups whose height kernels have positive finite-dimensional homology. The first controls which periodic groups can coexist; the second will keep their later joins quasiconvex in the ambient group.
Primitive presentations and actual trees
Definition 5.1 (Primitive data). Fix k ≥ 1 k\geq1 k ≥ 1 a i = t − i a t i a_i=t^{-i}at^i a i = t − i a t i
B 0 = { a 1 , … , a k − 1 } , P − = F ( a 0 , … , a k − 1 ) , P + = F ( a 1 , … , a k ) . B_0=\{a_1,\ldots,a_{k-1}\},\qquad P_-=F(a_0,\ldots,a_{k-1}),\qquad P_+=F(a_1,\ldots,a_k). B 0 = { a 1 , … , a k − 1 } , P − = F ( a 0 , … , a k − 1 ) , P + = F ( a 1 , … , a k ) . An empty B 0 B_0 B 0 p , q p,q p , q pr p / q ( v , w ) \operatorname{pr}_{p/q}(v,w) pr p / q ( v , w ) p / q p/q p / q
G = ⟨ a , t ∣ R ⟩ , R = pr p / q ( x , y ) or R = pr p / q ( x y , z ) . (9) G=\langle a,t\mid R\rangle,\qquad R=\operatorname{pr}_{p/q}(x,y)\ \text{or}\ R=\operatorname{pr}_{p/q}(xy,z). \tag*{(9)} G = ⟨ a , t ∣ R ⟩ , R = pr p / q ( x , y ) or R = pr p / q ( x y , z ) . (9) where
x ∈ P − ∖ F ( B 0 ) , y ∈ P + ∖ F ( B 0 ) , 1 ≠ z ∈ F ( B 0 ) x\in P_-\setminus F(B_0),\qquad y\in P_+\setminus F(B_0),\qquad1\ne z\in F(B_0) x ∈ P − ∖ F ( B 0 ) , y ∈ P + ∖ F ( B 0 ) , 1 = z ∈ F ( B 0 ) (the last choice is needed only in the second form). We assume that G G G [40 ] , Sections 3.1 and 5]. Explicitly, for the first form the family { ⟨ x ⟩ , ⟨ y ⟩ } \{\langle x\rangle,\langle y\rangle\} {⟨ x ⟩ , ⟨ y ⟩} a 0 , … , a k a_0,\ldots,a_k a 0 , … , a k p = q = 1 p=q=1 p = q = 1 x ′ y ′ x'y' x ′ y ′ x ′ , y ′ x',y' x ′ , y ′ F ( B 0 ) F(B_0) F ( B 0 ) ⟨ z ⟩ \langle z\rangle ⟨ z ⟩ p / q p/q p / q x ′ y ′ x'y' x ′ y ′
Let
h : G ⟶ Z , h ( a ) = 0 , h ( t ) = 1 , L = ker h . h:G\longrightarrow\mathbb{Z},\qquad h(a)=0,\qquad h(t)=1,\qquad L=\ker h. h : G ⟶ Z , h ( a ) = 0 , h ( t ) = 1 , L = ker h . The orientation can be reversed without changing any argument. Write H = ⟨ a 0 , … , a k ∣ R ⟩ H=\langle a_0,\ldots,a_k\mid R\rangle H = ⟨ a 0 , … , a k ∣ R ⟩ H H H P − P_- P − P + P_+ P + T 0 T_0 T 0
The refinement below is the graph-of-groups decomposition of [40 ] , Corollaries 3.10–3.11 and Section 5], with the edge embeddings kept explicit for later intersection arguments.
Lemma 5.2 (Refinement). There is an actual graph-of-groups decomposition
H = P − ∗ I Q ∗ J P + , I = F ( B 0 , x ) , J = F ( B 0 , y ) . (10) H=P_-\ast_I Q\ast_J P_+,\qquad I=F(B_0,x),\qquad J=F(B_0,y). \tag*{(10)} H = P − ∗ I Q ∗ J P + , I = F ( B 0 , x ) , J = F ( B 0 , y ) . (10) For nonzero integers d , n d,n d , n
Q = F ( B 0 , u ) , x = u d , y = u n , R = pr p / q ( x , y ) , Q = F ( B 0 , x ) ∗ ⟨ z ⟩ = ⟨ u d ⟩ ⟨ u ⟩ , y = x − 1 u n , R = pr p / q ( x y , z ) . \begin{aligned}
Q&=F(B_0,u),\qquad x=u^d,\qquad y=u^n,\qquad R=\operatorname{pr}_{p/q}(x,y),\\
Q&=F(B_0,x)\ast_{\langle z\rangle=\langle u^d\rangle}\langle u\rangle,\qquad y=x^{-1}u^n,\qquad R=\operatorname{pr}_{p/q}(xy,z).
\end{aligned} Q Q = F ( B 0 , u ) , x = u d , y = u n , R = pr p / q ( x , y ) , = F ( B 0 , x ) ∗ ⟨ z ⟩ = ⟨ u d ⟩ ⟨ u ⟩ , y = x − 1 u n , R = pr p / q ( x y , z ) . The second group has the same root-extension description with F ( B 0 , y ) F(B_0,y) F ( B 0 , y ) T 0 T_0 T 0 P P P T T T
Proof. A subgroup generated by a free factor F ( B 0 ) F(B_0) F ( B 0 ) F ( B 0 ) F(B_0) F ( B 0 ) I , J I,J I , J P P P
Amalgam normal form in F ( a 0 , … , a k ) = P − ∗ F ( B 0 ) P + F(a_0,\ldots,a_k)=P_-\ast_{F(B_0)}P_+ F ( a 0 , … , a k ) = P − ∗ F ( B 0 ) P + x , y x,y x , y x y , z xy,z x y , z u u u (10) and its displayed equations. The substitution y = x − 1 u n y=x^{-1}u^n y = x − 1 u n
Both extreme letters remain essential in the cyclically reduced relator. In the first form, nonzero powers of x x x y y y x y , z xy,z x y , z x , y x,y x , y z z z □ \square □
A row is a vertex or an edge midpoint ξ \xi ξ T T T X ξ X_\xi X ξ t i H t^iH t i H i i i P − P_- P − Q Q Q P + P_+ P + i + 1 2 i+\frac{1}{2} i + 2 1 i i i i − 1 2 i-\frac{1}{2} i − 2 1 P + P_+ P + P − P_- P − t i − 1 H t^{i-1}H t i − 1 H G G G h ( g x ) = h ( x ) + h ( g ) h(gx)=h(x)+h(g) h ( g x ) = h ( x ) + h ( g ) I I I J J J i + 1 4 i+\frac{1}{4} i + 4 1 i − 1 4 i-\frac{1}{4} i − 4 1 L L L T 0 T_0 T 0 L \ T 0 L\backslash T_0 L \ T 0 L L L T 0 T_0 T 0 T T T Figure 2 shows one part of the refinement and the resulting height convention.
Figure 2. A local part of the actual row refinement. Each coarse row is replaced by P − ∗ I Q ∗ J P + P_{-} *_I Q *_J P_{+} P − ∗ I Q ∗ J P + P P P I I I J J J h ( t ) = 1 h(t)=1 h ( t ) = 1 L L L
Lemma 5.3 (Row geometry and overlaps). All coarse and refined rows are hyperbolic, and their incident maps are quasi-isometric embeddings. Refined rows are quasiconvex in their containing coarse row H H H
(i) their ordinary intersection is F ( B 0 ) F(B_0) F ( B 0 ) F ( B 0 ) ∗ ⟨ w ⟩ F(B_0) * \langle w\rangle F ( B 0 ) ∗ ⟨ w ⟩ w w w
(ii) a conjugate intersection can be noncyclic only in the aligned opposite-slot double coset; distinct same-slot cosets intersect at most cyclically;
(iii) the overlap and its shifted copy have full-row-rank Fox matrices in their free sides. They are strongly inert there: for every finitely generated subgroup U U U P P P O ≤ P O\le P O ≤ P
∑ U g O ∈ U \ P / O max { rk ( U ∩ g O g − 1 ) − 1 , 0 } ≤ max { rk U − 1 , 0 } . \sum_{UgO\in U\backslash P/O} \max\{\operatorname{rk}(U\cap gOg^{-1})-1,0\}\le\max\{\operatorname{rk}U-1,0\}. U g O ∈ U \ P / O ∑ max { rk ( U ∩ g O g − 1 ) − 1 , 0 } ≤ max { rk U − 1 , 0 } . Proposition 5.7 proves the simultaneous subgroup-homology injection behind this inequality from the stated matrix property.
In a nonfree root row Q Q Q z z z
Proof. The one-relator group H H H G G G [52 ] . Its Magnus sides are quasiconvex [40 ] . Collins’s ordinary-intersection theorem gives (i) [15 ] , and his conjugate-intersection theorem gives (ii), including the case of equal Magnus subgroups [16 ] ; see also [52 ] . The common alphabet is a connected Magnus subgraph. For (iii), the Fox matrix keeps the identity columns for B 0 B_0 B 0 w w w
Each refined edge is finitely generated in a quasiconvex free P P P H H H H H H
If ∣ d ∣ = 1 |d| = 1 ∣ d ∣ = 1 z = v m z = v^m z = v m ∣ m ∣ > 1 |m| > 1 ∣ m ∣ > 1 ⟨ v , u ∣ v m = u d ⟩ \langle v,u \mid v^m = u^d \rangle ⟨ v , u ∣ v m = u d ⟩ Q Q Q C ∣ m ∣ ∗ C ∣ d ∣ C_{|m|} * C_{|d|} C ∣ m ∣ ∗ C ∣ d ∣ Z 2 \mathbb{Z}^2 Z 2 ⟨ z ⟩ \langle z\rangle ⟨ z ⟩
Coefficients and level homology
Notation 5.4. For U ≤ G U \le G U ≤ G D U D_U D U U ( U ) \mathcal{U}(U) U ( U ) [13 , 30 , 35 ] . We use natural subgroup embeddings, independence of distinct cosets over subgroup fields, and rational skew-Laurent extension over the kernel of a cyclic quotient [33 ] , Section 2.3 and Proposition 2.2]. Coset independence also holds for affiliated coefficients: clear finitely many coefficients to bounded subgroup operators and use their disjoint Fourier supports.
For V ≤ U V \le U V ≤ U H ∗ ( V ; D U ) H_*(V;D_U) H ∗ ( V ; D U ) D V D_V D V
β ( V ) = dim D V H 1 ( V ; D V ) . \beta(V) = \dim_{D_V} H_1(V;D_V). β ( V ) = dim D V H 1 ( V ; D V ) . This dimension is unchanged by overfield extension. For a nontrivial free group of finite rank r r r r − 1 r - 1 r − 1
Coefficients precede chain vectors and boundary matrices act on the right. Write d ( v ) d(v) d ( v ) v v v
d ( v w ) = d ( v ) + v d ( w ) , ∂ d ( v ) = v − 1 , Δ ( v ) = ( v − 1 ) − 1 d ( v ) ( v ≠ 1 ) . d(vw) = d(v) + v d(w), \qquad\partial d(v) = v - 1, \qquad\Delta(v) = (v - 1)^{-1}d(v) \quad(v \ne1). d ( v w ) = d ( v ) + v d ( w ) , ∂ d ( v ) = v − 1 , Δ ( v ) = ( v − 1 ) − 1 d ( v ) ( v = 1 ) . Thus ∂ Δ ( v ) = 1 \partial\Delta(v) = 1 ∂ Δ ( v ) = 1 Δ ( v ) − Δ ( w ) \Delta(v) - \Delta(w) Δ ( v ) − Δ ( w ) detects a noncyclic subgroup in ambient homology if some such difference from the latter subgroup has nonzero class.
Changing an orbit representative by s s s s s s s − 1 s^{-1} s − 1 s s s s v s − 1 svs^{-1} s v s − 1
Every noncyclic subgroup of a free group detects noncommensurable elements. Indeed the support realization of Section 6 places the normalized Fox chains near their respective cyclic axes. Those axes have bounded overlap when they are noncommensurable. Equality would give a finitely supported chain with boundary the mass at the identity, contrary to the zero total augmentation of a finite boundary.
The height-kernel presentation is
L = ⟨ a i ( i ∈ Z ) ∣ R i ( i ∈ Z ) ⟩ , (11) L = \langle a_i\ (i \in\mathbb{Z}) \mid R_i\ (i \in\mathbb{Z}) \rangle, \tag*{(11)} L = ⟨ a i ( i ∈ Z ) ∣ R i ( i ∈ Z )⟩ , (11) where R i R_i R i a i , … , a i + k a_i,\ldots,a_{i+k} a i , … , a i + k [43 ] , after collapsing its stable-letter tree.
Lemma 5.5 (Level basis). Over D L D_L D L the two-boundary in (11) is injective. Every interval of k k k consecutive a a a columns is a basis of C 1 / im ∂ 2 C_1/\operatorname{im}\partial_2 C 1 / im ∂ 2 The corresponding quotient for any refined row, after extension to D L D_L D L maps isomorphically to the global quotient. The incident edge-to-row maps are isomorphisms on these quotients, and
dim D L H 1 ( L ; D L ) = k − 1. \dim_{D_L} H_1(L;D_L)=k-1. dim D L H 1 ( L ; D L ) = k − 1. Proof. Adjoin u i u_i u i R i R_i R i x i = u i d x_i=u_i^d x i = u i d y i = u i n y_i=u_i^n y i = u i n u i d = z i u_i^d=z_i u i d = z i x i y i = u i n x_i y_i=u_i^n x i y i = u i n m ≠ 0 m\ne0 m = 0 u i u_i u i u i m u_i^m u i m D L D_L D L u i u_i u i a a a x i , y i x_i,y_i x i , y i
In any finite dependence between shifted two-boundaries, an outermost nonzero coefficient has an extreme entry which cannot cancel. Hence the two-boundary is injective. Conversely either extreme column can be eliminated using that shifted relation. Repeated elimination moves any finite chain into a prescribed k k k
For P P P ( B 0 , x ) (B_0,x) ( B 0 , x ) ( B 0 , y ) (B_0,y) ( B 0 , y ) u u u
The boundary of the resulting k k k D L D_L D L i s a u n i t f o r $ 1 ≠ v ∈ L is a unit for \$1\ne v\in L i s a u ni t f or $1 = v ∈ L
Proposition 5.6 (Restriction to row orbits). For U ≤ L U\le L U ≤ L and an actual level L \mathcal{L} L inclusion and the above coefficient transport give an injection
⨁ ξ ∈ U \ L H 1 ( U ∩ X ξ ; D U ) ⟶ H 1 ( U ; D U ) . (12) \bigoplus_{\xi\in U\backslash L} H_1(U\cap X_\xi;D_U)\longrightarrow H_1(U;D_U). \tag*{(12)} ξ ∈ U \ L ⨁ H 1 ( U ∩ X ξ ; D U ) ⟶ H 1 ( U ; D U ) . (12) For U U U in a vertex row, the analogous sum over one incident slot injects. The same assertions hold for coarse rows. The assertions hold in the induced chain modules before passing to an overfield.
Proof. Use
E U = D U ⊗ Q U Q L , E_U=D_U\otimes_{\mathbb{Q}U}\mathbb{Q}L, E U = D U ⊗ Q U Q L , embedded in D L D_L D L (12) [14 , 59 ] . Thus we must check that a row cycle bounding over E U E_U E U
For a root row keep the added columns u j u_j u j E U E_U E U Q Q Q i i i ( B 0 , x ) (B_0,x) ( B 0 , x ) x i x_i x i u i u_i u i j ≠ i j\ne i j = i x j y j = u j n x_jy_j=u_j^n x j y j = u j n D L D_L D L E U E_U E U u j u_j u j u j d = z j u_j^d=z_j u j d = z j i i i x i y i = u i n x_i y_i=u_i^n x i y i = u i n E U E_U E U x i x_i x i H H H
For a single slot replace L L L E U E_U E U
Proposition 5.7 (Path inertia). In T T T T 0 T_0 T 0 L L L H 1 ( L ; D L ) H_1(L;D_L) H 1 ( L ; D L ) G G G L L L
∑ γ m o d L max { rk Stab ( γ ) − 1 , 0 } ≤ k − 1. (13) \sum_{\gamma\bmod L} \max\{\operatorname{rk}\operatorname{Stab}(\gamma)-1,0\} \le k-1. \tag*{(13)} γ mod L ∑ max { rk Stab ( γ ) − 1 , 0 } ≤ k − 1. (13) The same joint-injection statement holds for continuations of any fixed additional shape inside any subgroup U U U U U U
Proof. Apply the slot rule with U U U H 1 ( U ; D U ) H_1(U;D_U) H 1 ( U ; D U ) T T T P P P
Begin at a free side and extend a monotone path one step in either direction. Only the aligned opposite-slot double coset can have a noncyclic intersection, by Lemma 5.3(ii). Its overlap O ≤ P O \le P O ≤ P U ≤ P U \le P U ≤ P D U ⊗ Q U Q P D_U \otimes_{Q_U} Q_P D U ⊗ Q U Q P U ∩ s O s − 1 U \cap sOs^{-1} U ∩ s O s − 1
This keeps the subgroup matrices throughout: restriction to a smaller subgroup merely decomposes the same induced modules again. If an independent finite family of the resulting cycles acquired an ambient relation, the trimming proof of Proposition 5.6 would reduce it to a relation at the preceding step. This supplies the restriction property used later for different double-coset orbit summands.
At each step, intersect quasiconvex subgroups in a hyperbolic row. Their intersections are finitely generated and quasiconvex, and only finitely many double cosets have nontrivial intersection. Iterating over a fixed shape gives the claimed finite lists. Each path group lies in a free edge group and is free. The joint injection and Lemma 5.5 prove the budget.
We will use more than the numerical budget. Inside a later path group, a predecessor already contributing its full first homology excludes a second noncyclic predecessor orbit. The induced-module statement also retains the distinction between different double cosets after restriction to a subgroup. These are the two forms of path inertia used in the trajectory construction of Section 9. If k = 1 k=1 k = 1
Lemma 5.8 (Cyclic-path dictionary). Fix a finite edge-containing path shape. In a reduced tree path fixed pointwise by a nontrivial element, the number of pairwise disjoint subsegments of that shape having cyclic stabilizer is bounded. Monotonicity is not required.
Proof. The fixed shape has finitely many nontrivial stabilizer G G G s s s c c c s s s ⟨ c ⟩ \langle c\rangle ⟨ c ⟩
Subtree gates and finite homology in a row
The path-homology estimate is now available. We turn to the second output of this section: recognizing when a periodic subgroup is quasiconvex in G G G
Lemma 5.9 (Subtree gate criterion). Let a torsion-free hyperbolic group J J J A ≤ J A \le J A ≤ J Y Y Y A \ Y A \backslash Y A \ Y A A A J J J
(i) ambient vertex rows are quasiconvex in J J J
(ii) every edge leaving Y Y Y A A A
This applies in particular to G G G T T T T 0 T_0 T 0
Proof. At every included vertex v v v A v = A ∩ J v A_v=A\cap J_v A v = A ∩ J v A A A W W W A A A A A A W W W
Inside an included vertex take coarse nearest-point projection to its orbit neighborhood. These projections agree across an included edge through their gates to its exact intersection subgroup. Indeed a geodesic in the attaching hull to a point of the intersection passes close to the vertex gate. It stays near the attaching hull, placing that gate in a bounded-neighborhood overlap. For two fixed subgroup cosets such an overlap is within bounded distance of their intersection orbit: properness leaves finitely many possible discrepancies of a close pair, and pairs with one fixed discrepancy form an orbit of the intersection subgroup. First-entry gates in the edge thus coarsely agree with both vertex gates. Quasi-isometric edge maps preserve this description by quasigeodesic stability. There are finitely many included attachment types, so the bounds are uniform and are measured in the intrinsic path metric of W W W
Each omitted tree component attaches across one edge. In case (i) the vertex subgroup at that edge is quasiconvex in J J J
For case (ii), consider an omitted attaching hull in the included boundary row. Its projection to the vertex-subgroup neighborhood has bounded diameter, or is within bounded distance of an orbit of their cyclic intersection. The alternatives are uniform. To see this, write the vertex subgroup as M M M g E gE g E E E E p ∈ M p \in M p ∈ M q ∈ g E q \in gE q ∈ g E p − 1 ∈ M p^{-1} \in M p − 1 ∈ M ( 1 , s ) (1,s) ( 1 , s ) s s s s E sE s E M M M M ∩ s E s − 1 M \cap sEs^{-1} M ∩ s E s − 1
A bounded projection extends by a constant. In an unbounded case choose a generator c c c f f f M M M J J J
∣ f ( x ) − f ( y ) ∣ ≤ C d J ( x , y ) + C . (14) \lvert f(x)-f(y)\rvert\le C d_J(x,y)+C. \tag*{(14)} ∣ f ( x ) − f ( y )∣ ≤ C d J ( x , y ) + C . (14) The inclusion of the boundary vertex row has a Cannon–Thurston extension [50 ] , since the splitting has finite quotient, hyperbolic vertex and edge groups, and quasi-isometrically embedded edge maps. Its periodic-axis endpoints map to the two distinct endpoints of c c c ∂ J \partial J ∂ J M 0 M_0 M 0 R 0 R_0 R 0 x , y x,y x , y M 0 M_0 M 0 B J ( 1 , R 0 ) B_J(1,R_0) B J ( 1 , R 0 ) n n n B J ( 1 , n ) B_J(1,n) B J ( 1 , n ) c c c B J ( 1 , n ) B_J(1,n) B J ( 1 , n ) M 0 , R 0 M_0,R_0 M 0 , R 0
Consequently an ambient geodesic from x x x y y y c j c^j c j c − j c^{-j} c − j j j j J J J (14) .
On the discrete attaching hull enlarge the constant so that f f f
f ~ ( z ) = inf x { f ( x ) + C d J ( x , z ) } . \widetilde{f}(z)=\inf_x\{f(x)+C d_J(x,z)\}. f ( z ) = x inf { f ( x ) + C d J ( x , z )} . Round its values to integers and send them to the same periodic orbit. This gives a coarse Lipschitz extension in the intrinsic target metric, agreeing with the attaching gate up to bounded error. Only the cyclic coordinate has been estimated in the ambient metric; undistortion of the boundary row was not assumed.
Combine these extensions with the compatible included-vertex gates. Across every ordinary or strip edge the image distance in W W W W W W A A A J J J
Lemma 5.10 (Finite homology in a refined row). Every subgroup M M M β ( M ) < ∞ \beta(M)<\infty β ( M ) < ∞ β ( M ) = 0 \beta(M)=0 β ( M ) = 0 M M M
Proof. For a free row this follows from the free-group rank formula. Consider a nonfree root row
Q = F ( B 0 , x ) ∗ ⟨ z ⟩ = ⟨ u d ⟩ ⟨ u ⟩ Q=F(B_0,x)*_{\langle z\rangle=\langle u^d\rangle}\langle u\rangle Q = F ( B 0 , x ) ∗ ⟨ z ⟩ = ⟨ u d ⟩ ⟨ u ⟩ and restrict its cyclic splitting to M M M H 1 ( M ; D M ) H_1(M;D_M) H 1 ( M ; D M )
At a cyclic free-base stabilizer, at most one incident edge has nontrivial intersection, and its maximal-cyclic property implies that this whole vertex stabilizer fixes that edge. At a noncyclic free-base stabilizer, quasiconvexity in the free base gives finitely many orbits of nontrivial edge intersections. Root-side vertices have valence ∣ d ∣ |d| ∣ d ∣
Collapse these components and split over the remaining trivial edges. Bass–Serre normal form expresses M M M M = 1 M=1 M = 1 M ≠ 1 M\ne1 M = 1 s s s r r r
∑ j = 1 s β ( M j ) + s + r − 1. \sum_{j=1}^{s}\beta(M_j)+s+r-1. j = 1 ∑ s β ( M j ) + s + r − 1. Infinitely many factors or free generators give infinite dimension, as follows either by finite subproducts or the degree-zero part of the tree sequence. Finite β ( M ) \beta(M) β ( M )
If β ( M ) = 0 \beta(M)=0 β ( M ) = 0
The free-base and cyclic vertices of the root splitting are quasiconvex in Q Q Q M M M M M M Q Q Q
Periodicity and the homology budget
Proposition 5.11 (Period rule). Suppose A ≤ G A\leq G A ≤ G h ( A ) ≠ 0 h(A)\ne0 h ( A ) = 0
S = A ∩ L , 0 < β ( S ) < ∞ . S=A\cap L,\qquad0<\beta(S)<\infty. S = A ∩ L , 0 < β ( S ) < ∞. At every actual level L \mathcal{L} L
⨁ ξ ∈ S \ L H 1 ( S ∩ X ξ ; D S ) ⟶ H 1 ( S ; D S ) . \bigoplus_{\xi\in S\backslash\mathcal{L}} H_1(S\cap X_\xi;D_S)\longrightarrow H_1(S;D_S). ξ ∈ S \ L ⨁ H 1 ( S ∩ X ξ ; D S ) ⟶ H 1 ( S ; D S ) . Both incident edge-level maps to a vertex level are isomorphisms. Every level sum therefore has dimension q X = β ( S ) q_X=\beta(S) q X = β ( S ) Y Y Y S ∩ X ξ S\cap X_\xi S ∩ X ξ S S S A A A A \ Y A\backslash Y A \ Y A A A G G G
Proof. Use D S D_S D S V j V_j V j j j j E j E_j E j E j → V j , V j + 1 E_j\to V_j,V_{j+1} E j → V j , V j + 1 C C C H 1 ( S ; D S ) H_1(S;D_S) H 1 ( S ; D S )
For an interval [ a , b ] [a,b] [ a , b ]
C [ a , b ] = ( ⨁ j = a b V j ) / im ( ⨁ j = a b − 1 E j ) . C[a,b]=\left(\bigoplus_{j=a}^{b}V_j\right)\big/\operatorname{im}\left(\bigoplus_{j=a}^{b-1}E_j\right). C [ a , b ] = ( j = a ⨁ b V j ) / im ( j = a ⨁ b − 1 E j ) . It injects into C C C
dim C [ a , b ] = dim V a + ∑ j = a b − 1 ( dim V j + 1 − dim E j ) . \dim C[a,b]=\dim V_a+\sum_{j=a}^{b-1}(\dim V_{j+1}-\dim E_j). dim C [ a , b ] = dim V a + j = a ∑ b − 1 ( dim V j + 1 − dim E j ) . Each summand is nonnegative. A positive-height element of A A A D S D_S D S dim C ≤ β ( S ) < ∞ \dim C\leq\beta(S)<\infty dim C ≤ β ( S ) < ∞ V j V_j V j C C C
The remaining part of the tree sequence is
K 0 = ker ( ⨁ edge orbits H 0 ( S e ; D S ) ⟶ ⨁ vertex orbits H 0 ( S v ; D S ) ) . K_0=\ker\left(\bigoplus_{\text{edge orbits}}H_0(S_e;D_S)\longrightarrow\bigoplus_{\text{vertex orbits}}H_0(S_v;D_S)\right). K 0 = ker edge orbits ⨁ H 0 ( S e ; D S ) ⟶ vertex orbits ⨁ H 0 ( S v ; D S ) . It is a quotient of H 1 ( S ; D S ) H_1(S;D_S) H 1 ( S ; D S ) K 0 K_0 K 0 K 0 = 0 K_0=0 K 0 = 0 H 1 ( S ; D S ) = C H_1(S;D_S)=C H 1 ( S ; D S ) = C
Take the actual path between rows with noncyclic stabilizer. Its edge chain with coefficient 1 1 1 H 0 H_0 H 0 K 0 K_0 K 0
By Lemma 5.10, all row intersections are finitely generated and quasiconvex; every noncyclic one contributes at least one dimension. The level bound and height periodicity give finitely many A A A A A A A A A Y Y Y A \ Y A\backslash Y A \ Y
At a boundary edge, the A A A S S S Y Y Y
Support and descent for division-ring coefficients
The closure construction requires two forms of algebraic coefficient control. Disjoint subgroup support patterns must give independent expressions. We also need descent from intersecting subgroup-field spans when the subgroup intersection is separable in both groups. Theorem 6.1 supplies these assertions. Section 7 supplies the separate localization argument for coefficients subject to height translation.
Throughout this section, J J J D J D_J D J Q \mathbb{Q} Q A ≤ J A \leq J A ≤ J D A D_A D A Q A \mathbb{Q}A Q A H ≤ J H \leq J H ≤ J χ : H ↠ Z \chi:H \twoheadrightarrow\mathbb{Z} χ : H ↠ Z K K K t ∈ H t \in H t ∈ H χ ( t ) = 1 \chi(t)=1 χ ( t ) = 1
D H = D K ( t ; Ad t ) ↪ D K ( ( t ; Ad t ) ) . (15) D_H=D_K(t;\operatorname{Ad}t)\hookrightarrow D_K((t;\operatorname{Ad}t)). \tag*{(15)} D H = D K ( t ; Ad t ) ↪ D K (( t ; Ad t )) . (15) The left side is the Ore division ring of skew Laurent polynomials, and the right side consists of series bounded below in degree. Distinct subgroup cosets are independent over the corresponding subgroup field. The normal-subgroup crossed-product assertion is [33 ] , Section 2.3 and Proposition 2.2; the compatible rational affiliated realization is [35 ] , Corollaries 6.2–6.3, extending the Hughes-free framework of [32 ] .
For arbitrary, possibly nonnormal A A A ∑ i d i g i = 0 \sum_i d_i g_i=0 ∑ i d i g i = 0 d i ∈ U ( A ) d_i \in\mathcal{U}(A) d i ∈ U ( A ) A g i Ag_i A g i b = ( 1 + ∑ i d i d i ∗ ) − 1 / 2 b=(1+\sum_i d_i d_i^*)^{-1/2} b = ( 1 + ∑ i d i d i ∗ ) − 1/2 b d i bd_i b d i b d i g i bd_i g_i b d i g i d i = 0 d_i=0 d i = 0
Our arguments concern algebraic rational expressions and do not assign Fourier coefficients to an arbitrary affiliated operator.
Permitted support patterns
A support pattern is a subset of J J J
P = A 1 g 1 A 2 g 2 ⋯ A m g m , A i ≤ J , g i ∈ J . P=A_1g_1A_2g_2\cdots A_mg_m,\qquad A_i\leq J,\quad g_i\in J. P = A 1 g 1 A 2 g 2 ⋯ A m g m , A i ≤ J , g i ∈ J . A product d 1 g 1 ⋯ d m g m d_1g_1\cdots d_mg_m d 1 g 1 ⋯ d m g m d i ∈ D A i d_i\in D_{A_i} d i ∈ D A i permitted on P P P S ⊆ J S\subseteq J S ⊆ J V ( S ) \mathcal{V}(S) V ( S ) Q \mathbb{Q} Q S S S V ( S ) g = V ( S g ) \mathcal{V}(S)g=\mathcal{V}(Sg) V ( S ) g = V ( S g )
Theorem 6.1 (Support and coset descent). The following assertions hold.
(i) If S ∩ T = ∅ S\cap T=\varnothing S ∩ T = ∅ then V ( S ) ∩ V ( T ) = 0 \mathcal{V}(S)\cap\mathcal{V}(T)=0 V ( S ) ∩ V ( T ) = 0
(ii) There is a Q \mathbb{Q} Q linear map
F : D J ⟶ Q J \mathcal{F}:D_J\longrightarrow\mathbb{Q}^{J} F : D J ⟶ Q J with F ( 1 ) = δ 1 \mathcal{F}(1)=\delta_1 F ( 1 ) = δ 1 commuting with every right group translation, such that v ∈ V ( S ) v\in\mathcal{V}(S) v ∈ V ( S ) implies supp F ( v ) ⊆ S \operatorname{supp}\mathcal{F}(v)\subseteq S supp F ( v ) ⊆ S
(iii) Let A , B ≤ J A,B\leq J A , B ≤ J and C = A ∩ B C=A\cap B C = A ∩ B If $At\cap Bs is empty, then D A t ∩ D B s = 0 D_{At}\cap D_{Bs}=0 D A t ∩ D B s = 0 Otherwise, for any x ∈ A t ∩ B s x\in At\cap Bs x ∈ A t ∩ B s
D A t ∩ D B s = D C x . (16) D_{At}\cap D_{Bs}=D_Cx. \tag*{(16)} D A t ∩ D B s = D C x . (16) If, in addition, C C C is separable in both A A A and $B, then
( ∑ g ∈ J D A g ) ∩ ( ∑ g ∈ J D B g ) = ∑ g ∈ J D C g . (17) \left(\sum_{g\in J}D_{Ag}\right)\cap\left(\sum_{g\in J}D_{Bg}\right)=\sum_{g\in J}D_{Cg}. \tag*{(17)} g ∈ J ∑ D A g ∩ g ∈ J ∑ D B g = g ∈ J ∑ D C g . (17) where each sum denotes its algebraic, finite-support span.
Applied entrywise to a finite row v v v
F ( v R ) = F ( v ) R \mathcal{F}(vR)=\mathcal{F}(v)R F ( v R ) = F ( v ) R for every compatible finite matrix R R R Q J \mathbb{Q}J Q J
Indeed each matrix entry acts by a finite sum of right group translations. Thus the same scalar-function map preserves the Fox boundary and divergence equations used in the closure construction. This consequence uses linearity and right equivariance; it requires no multiplicativity for division-ring coefficients.
We first make the complexity argument precise. The two descent arguments will use the same coefficient extraction, but have different conclusions: disjoint supports give vanishing, while intersecting single cosets give a smaller field.
A hereditary complexity and coefficient extraction
For predecessors of coefficient descent by rational-expression complexity, see Dicks–Herbera–Sánchez [18 ] , Sections 3–6, especially Theorem 6.3 and Jaikin-Zapirain–López-Álvarez [35 ] , Proposition 5.1. Here we combine an explicit ordinal complexity with bookkeeping for the prescribed subgroup support patterns.
The difficulty in an induction is inversion. A coefficient of an inverse series is a finite sum of products, but neither the lengths of those products nor the number of their displayed terms has a uniform bound. We therefore use two ordinal layers: the first makes simplifying an inverse operand outweigh any finite increase in product length, and the second makes simplifying a term outweigh any finite increase in the number of terms. We define them on the displayed expressions, retaining terms even when their values cancel.
A displayed rational expression can be normalized recursively as a finite sum of terms
q P 1 − 1 ⋯ P s − 1 g , q ∈ Q ∖ { 0 } , g ∈ J . (18) qP_1^{-1}\cdots P_s^{-1}g,\qquad q\in\mathbb{Q}\setminus\{0\},\qquad g\in J. \tag*{(18)} q P 1 − 1 ⋯ P s − 1 g , q ∈ Q ∖ { 0 } , g ∈ J . (18) Each P i P_i P i 1 1 1 g g g g P − 1 = ( g P g − 1 ) − 1 g gP^{-1}=(gPg^{-1})^{-1}g g P − 1 = ( g P g − 1 ) − 1 g
The hereditary offsets of an expression are all final group elements appearing at any depth of its displayed expression tree. For an inverse occurrence P − 1 P^{-1} P − 1 H ( P ) H(P) H ( P ) P P P
H ( P 1 ) ⋯ H ( P s ) g . H(P_1)\cdots H(P_s)g. H ( P 1 ) ⋯ H ( P s ) g . When the original factors have prescribed subgroup-field patterns, normalize them separately before multiplying. Every resulting term pattern is then contained in its original prescribed pattern. Indeed all offsets used to represent an element of D A D_A D A A A A
Write ⊕ \oplus ⊕ path weight and complexity recursively by
w ( q P 1 − 1 ⋯ P s − 1 g ) = ⨁ i = 1 s ω c ( P i ) , (19) w(qP_1^{-1}\cdots P_s^{-1}g)=\bigoplus_{i=1}^{s}\omega^{c(P_i)}, \tag*{(19)} w ( q P 1 − 1 ⋯ P s − 1 g ) = i = 1 ⨁ s ω c ( P i ) , (19) c ( E ) = ⨁ T a displayed term of E ω w ( T ) . (20) c(E)=\bigoplus_{T\text{ a displayed term of }E}\omega^{w(T)}. \tag*{(20)} c ( E ) = T a displayed term of E ⨁ ω w ( T ) . (20) An empty sum has complexity zero. Every complexity here is a finite hereditary ordinal below ε 0 \varepsilon_0 ε 0 r r r r r r
We shall use two elementary properties of this measure. Replacing an occurrence of ω α \omega^\alpha ω α ω β i \omega^{\beta_i} ω β i β i < α \beta_i < \alpha β i < α η \eta η η \eta η
Relative to a character χ \chi χ P − 1 P^{-1} P − 1 P P P nonhomogeneous when χ \chi χ P P P H ( P ) H(P) H ( P ) 1 + t − t 1+t-t 1 + t − t χ ( t ) ≠ 0 \chi(t)\ne0 χ ( t ) = 0 χ \chi χ
Lemma 6.2 (Coefficient extraction invariant). Let finitely many normalized rational expressions have hereditary offsets in a finitely generated subgroup H H H χ : H ↠ Z \chi:H \twoheadrightarrow\mathbb{Z} χ : H ↠ Z K K K t t t D K D_K D K
(a) If a summand is permitted on S S S n n n
( S ∩ χ − 1 ( n ) ) t − n . \left(S \cap\chi^{-1}(n)\right)t^{-n}. ( S ∩ χ − 1 ( n ) ) t − n . Here S S S H H H
(b) Coefficient extraction does not increase the complexity of an arbitrary top expression. If an inverse occurrence is nonhomogeneous, the decrease is strict. In particular, for a normalized operand P P P χ \chi χ c ( P ) c(P) c ( P )
(c) For a collection of top expressions, assume at least one represented top offset equals 1 1 1 χ \chi χ
(d) If the input is represented in one field coset D A a D_{Aa} D A a A A A n n n D A ∩ K D_{A\cap K} D A ∩ K ( A a ∩ χ − 1 ( n ) ) t − n (Aa\cap\chi^{-1}(n))t^{-n} ( A a ∩ χ − 1 ( n )) t − n A ∩ K A\cap K A ∩ K
Proof. We construct the coefficients by induction on the expression trees. An inverse operand whose hereditary offsets are all killed by χ \chi χ P P P H ( P ) H(P) H ( P ) χ ( H ( P ) ) = d Z \chi(H(P))=d\mathbb{Z} χ ( H ( P )) = d Z a ∈ H ( P ) a\in H(P) a ∈ H ( P ) χ ( a ) = d > 0 \chi(a)=d>0 χ ( a ) = d > 0 P P P a a a
( 1 + R ) − 1 = ∑ j ≥ 0 ( − R ) j , (1+R)^{-1}=\sum_{j\geq0}(-R)^j, ( 1 + R ) − 1 = j ≥ 0 ∑ ( − R ) j , where R R R P P P
For completeness, the complexity comparison can be proved at the same time by structural induction. A normalized nonhomogeneous operand P P P P P P c ( P ) c(P) c ( P ) P − 1 P^{-1} P − 1 c ( P ) c(P) c ( P ) c ( P ) c(P) c ( P ) P − 1 P^{-1} P − 1 ω c ( P ) \omega^{c(P)} ω c ( P )
This proves (b). For (c), retain the tags recording each original term. If some inverse occurrence is nonhomogeneous, every resulting term coming from that occurrence has smaller path weight, while unchanged terms contribute at most once at any degree. If all inverse occurrences are homogeneous, the top offset 1, together with nontriviality on all hereditary offsets, forces another top degree. Every coefficient then omits a top term. This proves (c), also for a joint collection whose individual expressions need not have a top offset 1.
It remains to check that this construction respects the designated supports rather than just the field identities. In the local expansion at P − 1 P^{-1} P − 1 H ( P ) H(P) H ( P ) n n n H ( P ) ∩ K H(P)\cap K H ( P ) ∩ K a m = k m t d m a^m=k_m t^{d_m} a m = k m t d m k m ∈ K k_m\in K k m ∈ K k m k_m k m t t t t − n t^{-n} t − n (19) –(20) . Conjugations and final-offset changes preserve the measure. This proves (a) and its compatibility with (b) and (c). For (d), all the local transporters before the final offset lie in A A A A A A A ∩ K A\cap K A ∩ K D A D_A D A A A A
Disjoint supports and the scalar-function map
Proof of Theorem 6.1(i)–(ii). Suppose v ∈ V ( S ) ∩ V ( T ) v\in\mathcal{V}(S)\cap\mathcal{V}(T) v ∈ V ( S ) ∩ V ( T ) S ∩ T S\cap T S ∩ T H H H
If H = 1 H=1 H = 1 { 1 } \{1\} { 1 } χ : H → Z \chi:H\to\mathbb{Z} χ : H → Z K K K (15) makes both original values zero as well. Well-foundedness of the ordinal measure completes the induction.
Set W = V ( J ∖ { 1 } ) W = \mathcal{V}(J \setminus\{1\}) W = V ( J ∖ { 1 }) W ∩ Q 1 = 0 W \cap\mathbb{Q}1 = 0 W ∩ Q 1 = 0 W + Q 1 W + \mathbb{Q}1 W + Q 1 ℓ ( w + q ) = q \ell(w + q) = q ℓ ( w + q ) = q Q \mathbb{Q} Q D J D_J D J
F ( v ) ( g ) = ℓ ( v g − 1 ) . (21) \mathcal{F}(v)(g) = \ell(vg^{-1}). \tag*{(21)} F ( v ) ( g ) = ℓ ( v g − 1 ) . (21) Then F ( 1 ) = δ 1 \mathcal{F}(1) = \delta_1 F ( 1 ) = δ 1 g ∉ S g \notin S g ∈ / S v ∈ V ( S ) v \in\mathcal{V}(S) v ∈ V ( S ) v g − 1 ∈ V ( S g − 1 ) ⊆ W vg^{-1} \in\mathcal{V}(Sg^{-1}) \subseteq W v g − 1 ∈ V ( S g − 1 ) ⊆ W
F ( v h ) ( g ) = ℓ ( v h g − 1 ) = F ( v ) ( g h − 1 ) , \mathcal{F}(vh)(g) = \ell(vhg^{-1}) = \mathcal{F}(v)(gh^{-1}), F ( v h ) ( g ) = ℓ ( v h g − 1 ) = F ( v ) ( g h − 1 ) , which is precisely equivariance for right group multiplication. This proves all support restrictions simultaneously. No multiplicativity of F \mathcal{F} F
Rational descent from a smaller coefficient field
We isolate the rational-series step, including its handedness.
Lemma 6.3 (Rational descent). Let E ⊆ F E \subseteq F E ⊆ F u u u F F F E E E F ( u ) F(u) F ( u )
S = ∑ n u n f n , f n ∈ E , S = \sum_n u^n f_n,\qquad f_n \in E, S = n ∑ u n f n , f n ∈ E , then S ∈ E ( u ) S \in E(u) S ∈ E ( u )
Proof. Choose a nonzero right Laurent-polynomial denominator P = ∑ j ∈ I u j p j P = \sum_{j \in I} u^j p_j P = ∑ j ∈ I u j p j I I I p j ∈ F p_j \in F p j ∈ F S P SP S P n n n u n u^n u n
∑ j ∈ I ( u − j f n − j u j ) p j = 0. (22) \sum_{j \in I} (u^{-j} f_{n-j} u^j)p_j = 0. \tag*{(22)} j ∈ I ∑ ( u − j f n − j u j ) p j = 0. (22) Every parenthesized coefficient belongs to E E E ( b γ ) γ (b_\gamma)_\gamma ( b γ ) γ F F F E E E p j = ∑ γ e j γ b γ p_j = \sum_\gamma e_{j\gamma}b_\gamma p j = ∑ γ e j γ b γ ( e j γ ) j ∈ I (e_{j\gamma})_{j \in I} ( e j γ ) j ∈ I (22) simultaneously. Since P ≠ 0 P \ne0 P = 0 P ′ = ∑ j ∈ I u j e j γ ≠ 0 P' = \sum_{j \in I} u^j e_{j\gamma} \ne0 P ′ = ∑ j ∈ I u j e j γ = 0
The product S P ′ SP' S P ′ E E E E E E S = ( S P ′ ) ( P ′ ) − 1 ∈ E ( u ) S = (SP')(P')^{-1} \in E(u) S = ( S P ′ ) ( P ′ ) − 1 ∈ E ( u )
Intersections of field cosets
Proof of Theorem 6.1(iii), single cosets. The empty-intersection assertion follows from part (i). Suppose A t ∩ B s = C x At \cap Bs = Cx A t ∩ B s = C x 0 ≠ v ∈ D A t ∩ D B s 0 \ne v \in D_{At} \cap D_{Bs} 0 = v ∈ D A t ∩ D B s v v v A A A B B B H H H
Intersect the two permitted cosets with H H H A ∩ H A \cap H A ∩ H B ∩ H B \cap H B ∩ H H H H H H H A A A B B B C C C H H H H = 1 H=1 H = 1 χ : H ↠ Z \chi:H\mathbin{\twoheadrightarrow}\mathbb{Z} χ : H ↠ Z K K K
For each degree n n n A ∩ K A\cap K A ∩ K B ∩ K B\cap K B ∩ K K K K A ∩ B ∩ K A\cap B\cap K A ∩ B ∩ K C x Cx C x D C ∩ K D_{C\cap K} D C ∩ K
Choose a common point x x x H H H v v v C C C χ ( C ) = 0 \chi(C)=0 χ ( C ) = 0 D C ∩ K = D C D_{C\cap K}=D_C D C ∩ K = D C χ ( C ) = d Z \chi(C)=d\mathbb{Z} χ ( C ) = d Z d > 0 d>0 d > 0 u ∈ C u\in C u ∈ C d d d E = D C ∩ K E=D_{C\cap K} E = D C ∩ K F = D K F=D_K F = D K
v x − 1 = ∑ m u m f m , f m ∈ E , (23) vx^{-1}=\sum_m u^m f_m,\qquad f_m\in E, \tag*{(23)} v x − 1 = m ∑ u m f m , f m ∈ E , (23) and conjugation by u u u E E E
We must also check that (23) is rational over F ( u ) F(u) F ( u ) F ( t ) F(t) F ( t ) H d = χ − 1 ( d Z ) H_d=\chi^{-1}(d\mathbb{Z}) H d = χ − 1 ( d Z ) d d d H H H F ( u ) F(u) F ( u )
D H = ⨁ r = 0 d − 1 D H d t r . D_H=\bigoplus_{r=0}^{d-1}D_{H_d}t^r. D H = r = 0 ⨁ d − 1 D H d t r . The Laurent series of the r r r r r r d d d (23) has only zero-congruence degrees, injectivity of Laurent expansion makes all other summands vanish. Hence v x − 1 ∈ F ( u ) vx^{-1}\in F(u) v x − 1 ∈ F ( u ) v x − 1 ∈ E ( u ) = D C vx^{-1}\in E(u)=D_C v x − 1 ∈ E ( u ) = D C C ≤ A , B C\leq A,B C ≤ A , B (16) in the original frame. If a restricted ambient subgroup was used, its common subgroup is contained in the original C C C
Finite spans and separability
Proof of Theorem 6.1(iii), finite spans. One inclusion in (17) is immediate. For the other, write a common element as
v = ∑ i = 1 r a i t i = ∑ j = 1 s b j s j , a i ∈ D A , b j ∈ D B . v=\sum_{i=1}^{r}a_i t_i=\sum_{j=1}^{s}b_j s_j,\qquad a_i\in D_A,\quad b_j\in D_B. v = i = 1 ∑ r a i t i = j = 1 ∑ s b j s j , a i ∈ D A , b j ∈ D B . Merge repeated cosets on each side, so the original A t i At_i A t i B s j Bs_j B s j A t i ∩ B s j = C x i j At_i\cap Bs_j=Cx_{ij} A t i ∩ B s j = C x ij
For a fixed i i i C x i j Cx_{ij} C x ij C x i k Cx_{ik} C x ik x i j x i k − 1 ∈ A ∖ C x_{ij}x_{ik}^{-1}\in A\setminus C x ij x ik − 1 ∈ A ∖ C A 0 ≤ A A_0\leq A A 0 ≤ A C C C A 0 A_0 A 0 A t i At_i A t i B B B A 0 A_0 A 0 c x i j ( c ′ x i k ) − 1 ∈ A 0 cx_{ij}(c'x_{ik})^{-1}\in A_0 c x ij ( c ′ x ik ) − 1 ∈ A 0 c , c ′ ∈ C c,c'\in C c , c ′ ∈ C x i j x i k − 1 ∈ A 0 x_{ij}x_{ik}^{-1}\in A_0 x ij x ik − 1 ∈ A 0 B 0 ≤ B B_0\leq B B 0 ≤ B C C C
These two choices already give the simultaneous refinement required. Since A 0 ∩ B = C A_0\cap B=C A 0 ∩ B = C A 0 A_0 A 0 B B B C C C C ≤ B 0 C \le B_0 C ≤ B 0 B 0 B_0 B 0 A 0 ∩ B 0 = C A_0 \cap B_0 = C A 0 ∩ B 0 = C
We use the finite-index field decompositions
D A = ⨁ a ∈ A 0 \ A D A 0 a , D B = ⨁ b ∈ B 0 \ B D B 0 b . D_A = \bigoplus_{a\in A_0\backslash A} D_{A_0}a,\qquad D_B = \bigoplus_{b\in B_0\backslash B} D_{B_0}b. D A = a ∈ A 0 \ A ⨁ D A 0 a , D B = b ∈ B 0 \ B ⨁ D B 0 b . For clarity, these do not require the chosen subgroups to be normal. One may first pass to a finite-index normal core: its finite crossed span in the overfield is a finite-dimensional domain over its division ring, hence a division ring, and so equals the larger subgroup field. Grouping the resulting summands by the original subgroup cosets gives the displayed formulas.
Refine the two expressions for v v v D A 0 D_{A_0} D A 0 D B 0 D_{B_0} D B 0 (16) , their common value belongs to D C x D_Cx D C x (17) .
Semilinear localization of affiliated coefficients
Write N ( J ) \mathcal{N}(J) N ( J ) J J J U ( J ) \mathcal{U}(J) U ( J )
Lemma 7.1 (Semilinear finite-dimensional support). Let G G G h : G → Z h:G\to\mathbb{Z} h : G → Z L = ker h L=\ker h L = ker h D ≤ G D\le G D ≤ G S = D ∩ L S=D\cap L S = D ∩ L g ∈ G g\in G g ∈ G g D ∈ D g_D\in D g D ∈ D
h ( g ) = h ( g D ) ≠ 0. h(g)=h(g_D)\ne0. h ( g ) = h ( g D ) = 0. Suppose that the division fields in use are embedded in the affiliated group algebras, and that, for an integer r ≥ 1 r\ge1 r ≥ 1 λ ∈ D L 1 × r \lambda\in D_L^{1\times r} λ ∈ D L 1 × r
g λ g D − 1 = λ B , B ∈ GL r ( D S ) . g\lambda g_D^{-1}=\lambda B,\qquad B\in\operatorname{GL}_r(D_S). g λ g D − 1 = λ B , B ∈ GL r ( D S ) . Then
λ ∈ ∑ x S ∈ P x U ( S ) 1 × r , P = { x S : x ∈ L , g m ∈ x D x − 1 for some m ∈ Z ∖ { 0 } } . (24) \lambda\in\sum_{xS\in\mathcal{P}}x\mathcal{U}(S)^{1\times r},\qquad\mathcal{P}=\{xS:x\in L,\ g^m\in xDx^{-1}\text{ for some }m\in\mathbb{Z}\setminus\{0\}\}. \tag*{(24)} λ ∈ x S ∈ P ∑ x U ( S ) 1 × r , P = { x S : x ∈ L , g m ∈ x D x − 1 for some m ∈ Z ∖ { 0 }} . (24) The set P \mathcal{P} P λ \lambda λ B B B (7.1) .
Here is the normalization behind the proof. Put α = Ad g D \alpha=\operatorname{Ad}g_D α = Ad g D w ↦ λ w w\mapsto\lambda w w ↦ λ w S S S P P P S S S
P = B α ( P ) B ∗ . P=B\alpha(P)B^*. P = B α ( P ) B ∗ . Then
Z = λ P 1 / 2 , u = P − 1 / 2 B α ( P ) 1 / 2 Z=\lambda P^{1/2},\qquad u=P^{-1/2}B\alpha(P)^{1/2} Z = λ P 1/2 , u = P − 1/2 B α ( P ) 1/2 satisfy u u ∗ = u ∗ u = 1 uu^* = u^*u = 1 u u ∗ = u ∗ u = 1 g Z g D − 1 = Z u gZg_D^{-1} = Zu g Z g D − 1 = Z u g Z Z ∗ g − 1 = Z Z ∗ gZZ^*g^{-1} = ZZ^* g Z Z ∗ g − 1 = Z Z ∗ g g g Z Z Z S S S ℓ 2 ( L ) \ell^2(L) ℓ 2 ( L ) r r r D D D
The invariant-form strategy parallels the bounded unitarization argument of Sz.-Nagy and Dixmier [19 ] , Section 5, Theorem 6. The positive mean constructed below supplies P P P N ( S ) \mathcal{N}(S) N ( S )
The affiliated conclusion in (24) is sufficient for Proposition 10.5. There the Bass–Serre homology injection is first established over a subgroup division field. Extending scalars to its affiliated algebra preserves that injection and its nonzero classes: choosing a vector-space basis identifies the scalar extension with a direct sum of copies of the nonzero affiliated algebra. Injectivity into ambient homology still requires the separate comparison using actual row chains and coset independence in that proposition.
Measure estimates and a projection inequality
Let ( M , τ ) (\mathcal{M},\tau) ( M , τ ) unnormalized traces
τ r ( ( a i j ) ) = ∑ j = 1 r τ ( a j j ) . \tau_r((a_{ij})) = \sum_{j=1}^{r}\tau(a_{jj}). τ r (( a ij )) = j = 1 ∑ r τ ( a j j ) . In particular, for a bounded rectangular matrix a a a τ r ( a ∗ a ) = τ k ( a a ∗ ) \tau_r(a^*a) = \tau_k(aa^*) τ r ( a ∗ a ) = τ k ( a a ∗ )
A family A \mathcal{A} A tight if
lim K → ∞ sup a ∈ A τ ( 1 ( K , ∞ ) ( ∣ a ∣ ) ) = 0. \lim_{K\to\infty}\sup_{a\in\mathcal{A}}\tau\left(1_{(K,\infty)}(|a|)\right)=0. K → ∞ lim a ∈ A sup τ ( 1 ( K , ∞ ) ( ∣ a ∣ ) ) = 0. For rectangular operators the trace is taken in the source corner; the nonzero spectral distributions of ∣ a ∣ |a| ∣ a ∣ ∣ a ∗ ∣ |a^*| ∣ a ∗ ∣ a j → 0 a_j \to0 a j → 0 ε , η > 0 \varepsilon,\eta>0 ε , η > 0 q q q τ ( 1 − q ) < η \tau(1-q)<\eta τ ( 1 − q ) < η ∥ a j q ∥ < ε \lVert a_jq\rVert<\varepsilon ∥ a j q ∥ < ε
Lemma 7.2 (Measure calculus). The affiliated algebra of a finite von Neumann algebra is complete for convergence in measure. Products of tight families are tight. If a j → 0 a_j\to0 a j → 0 b j , c j b_j,c_j b j , c j b j a j c j → 0 b_ja_jc_j\to0 b j a j c j → 0
Positive square roots are continuous in measure on positive affiliated operators, and inversion is continuous in measure on positive affiliated units. A uniformly norm-bounded family that tends to zero in measure tends to zero in L 1 ( τ ) L^1(\tau) L 1 ( τ )
Proof. The complete topological ∗ * ∗ [21 ] , Sections 1.4–1.5, Lemma 2.5 and Lemma 3.1. For clarity, the estimates that provide all the uniformity we need are
μ s + t ( a b ) ≤ μ s ( a ) μ t ( b ) , μ s + t ( a + b ) ≤ μ s ( a ) + μ t ( b ) , s , t > 0 , \mu_{s+t}(ab) \le\mu_s(a)\mu_t(b), \qquad\mu_{s+t}(a+b) \le\mu_s(a)+\mu_t(b), \qquad s,t>0, μ s + t ( ab ) ≤ μ s ( a ) μ t ( b ) , μ s + t ( a + b ) ≤ μ s ( a ) + μ t ( b ) , s , t > 0 , where
μ s ( a ) = inf { ∥ a q ∥ : q a projection , τ ( 1 − q ) ≤ s } . \mu_s(a)=\inf\{\lVert aq\rVert: q\text{ a projection},\ \tau(1-q)\le s\}. μ s ( a ) = inf {∥ a q ∥ : q a projection , τ ( 1 − q ) ≤ s } . Their proofs cut the two operators to bounded parts and add the trace codimensions; polar decomposition identifies the trace codimensions of the source and range cuts. Tightness says precisely that sup a ∈ A μ s ( a ) < ∞ \sup_{a\in A}\mu_s(a)<\infty sup a ∈ A μ s ( a ) < ∞ s > 0 s>0 s > 0 μ s ( a j ) → 0 \mu_s(a_j)\to0 μ s ( a j ) → 0 s > 0 s>0 s > 0
We recall how to obtain the stated functional calculus without any norm bounds on the original operators. For positive a j → a a_j\to a a j → a
( 1 + a j ) − 1 − ( 1 + a ) − 1 = ( 1 + a j ) − 1 ( a − a j ) ( 1 + a ) − 1 ⟶ 0 (1+a_j)^{-1}-(1+a)^{-1}=(1+a_j)^{-1}(a-a_j)(1+a)^{-1}\longrightarrow0 ( 1 + a j ) − 1 − ( 1 + a ) − 1 = ( 1 + a j ) − 1 ( a − a j ) ( 1 + a ) − 1 ⟶ 0 in measure. On bounded self-adjoint contractions, polynomial approximation gives continuity of continuous functional calculus in measure. Apply this to the positive contractions q j = ( 1 + a j ) − 1 q_j=(1+a_j)^{-1} q j = ( 1 + a j ) − 1 q = ( 1 + a ) − 1 q=(1+a)^{-1} q = ( 1 + a ) − 1 q q q 1 1 1 η > 0 \eta>0 η > 0 b < 1 b<1 b < 1 μ η / 2 ( q ) < b \mu_{\eta/2}(q)<b μ η /2 ( q ) < b μ η ( q j ) ≤ μ η / 2 ( q ) + μ η / 2 ( q j − q ) \mu_\eta(q_j)\le\mu_{\eta/2}(q)+\mu_{\eta/2}(q_j-q) μ η ( q j ) ≤ μ η /2 ( q ) + μ η /2 ( q j − q ) q j q_j q j 1 1 1 η \eta η j j j 1 − q j 1-q_j 1 − q j 0 0 0 q q q a a a ( q − 1 − 1 ) 1 / 2 (q^{-1}-1)^{1/2} ( q − 1 − 1 ) 1/2 ( q − 1 − 1 ) − 1 (q^{-1}-1)^{-1} ( q − 1 − 1 ) − 1 [ 0 , 1 ] [0,1] [ 0 , 1 ]
Finally, if ∥ z j ∥ ≤ C \lVert z_j\rVert\le C ∥ z j ∥ ≤ C
∥ z j ∥ 1 ≤ ε τ ( 1 ) + C τ ( 1 ( ε , ∞ ) ( ∣ z j ∣ ) ) . \lVert z_j\rVert_1\le\varepsilon\tau(1)+C\tau\bigl(1_{(\varepsilon,\infty)}(|z_j|)\bigr). ∥ z j ∥ 1 ≤ ε τ ( 1 ) + C τ ( 1 ( ε , ∞ ) ( ∣ z j ∣ ) ) . First use convergence in measure and then let ε \varepsilon ε
A positive affiliated unit in a corner is a positive affiliated operator whose support is the identity of that corner; its inverse is allowed to be unbounded. For two such units define
F ( P , X ) = P 1 / 2 ( P + X ) − 1 P 1 / 2 = ( 1 + P − 1 / 2 X P − 1 / 2 ) − 1 . F(P,X)=P^{1/2}(P+X)^{-1}P^{1/2}
=\left(1+P^{-1/2}XP^{-1/2}\right)^{-1}. F ( P , X ) = P 1/2 ( P + X ) − 1 P 1/2 = ( 1 + P − 1/2 X P − 1/2 ) − 1 . This is a bounded positive contraction.
Lemma 7.3 (Spectral-projection comparison). Let T , X T,X T , X M \mathcal{M} M t > 0 t>0 t > 0 p + = 1 ( t , ∞ ) ( T ) p_+=1_{(t,\infty)}(T) p + = 1 ( t , ∞ ) ( T ) p − = 1 ( 0 , t ) ( T ) p_-=1_{(0,t)}(T) p − = 1 ( 0 , t ) ( T )
τ ( p + F ( T , X ) ) ≥ τ ( p + F ( t 1 , X ) ) , (25) \tau\bigl(p_+F(T,X)\bigr)\ge\tau\bigl(p_+F(t1,X)\bigr), \tag*{(25)} τ ( p + F ( T , X ) ) ≥ τ ( p + F ( t 1 , X ) ) , (25) τ ( p − F ( T , X ) ) ≤ τ ( p − F ( t 1 , X ) ) . (26) \tau\bigl(p_-F(T,X)\bigr)\le\tau\bigl(p_-F(t1,X)\bigr). \tag*{(26)} τ ( p − F ( T , X ) ) ≤ τ ( p − F ( t 1 , X ) ) . (26) Proof. First assume that T , X T,X T , X p = p + p=p_+ p = p + q = 1 − p q=1-p q = 1 − p p + q p+q p + q
T = ( A 0 0 B ) , X = ( X 11 X 12 X 21 X 22 ) . T=\begin{pmatrix} A&0\\0&B\end{pmatrix},\qquad
X=\begin{pmatrix} X_{11}&X_{12}\\X_{21}&X_{22}\end{pmatrix}. T = ( A 0 0 B ) , X = ( X 11 X 21 X 12 X 22 ) . Here A ≥ t p A\ge tp A ≥ tp B ≤ t q B\le tq B ≤ tq
K B = X 11 − X 12 ( B + X 22 ) − 1 X 21 , K t = X 11 − X 12 ( t q + X 22 ) − 1 X 21 K_B=X_{11}-X_{12}(B+X_{22})^{-1}X_{21},\qquad
K_t=X_{11}-X_{12}(tq+X_{22})^{-1}X_{21} K B = X 11 − X 12 ( B + X 22 ) − 1 X 21 , K t = X 11 − X 12 ( tq + X 22 ) − 1 X 21 satisfy K B ≤ K t K_B\le K_t K B ≤ K t
τ ( p F ( T , X ) ) = τ ( A ( A + K B ) − 1 ) . \tau\bigl(pF(T,X)\bigr)=\tau\bigl(A(A+K_B)^{-1}\bigr). τ ( pF ( T , X ) ) = τ ( A ( A + K B ) − 1 ) . For positive invertible A , K A,K A , K p M p p\mathcal{M}p p M p
ψ ( A , K ) = τ ( A ( A + K ) − 1 ) . \psi(A,K)=\tau\left(A(A+K)^{-1}\right). ψ ( A , K ) = τ ( A ( A + K ) − 1 ) . This function is increasing in A A A
ψ ( A , K ) = τ ( p ) − τ ( K 1 / 2 ( A + K ) − 1 K 1 / 2 ) , \psi(A,K)=\tau(p)-\tau\left(K^{1/2}(A+K)^{-1}K^{1/2}\right), ψ ( A , K ) = τ ( p ) − τ ( K 1/2 ( A + K ) − 1 K 1/2 ) , and is decreasing in K K K A 1 / 2 A^{1/2} A 1/2
ψ ( A , K B ) ≥ ψ ( t p , K B ) ≥ ψ ( t p , K t ) = τ ( p F ( t 1 , X ) ) , \psi(A,K_B)\geq\psi(tp,K_B)\geq\psi(tp,K_t)=\tau\left(pF(t1,X)\right), ψ ( A , K B ) ≥ ψ ( tp , K B ) ≥ ψ ( tp , K t ) = τ ( pF ( t 1 , X ) ) , which proves (25) . For p = p − p=p_- p = p − A ≤ t p A\leq tp A ≤ tp B ≥ t q B\geq tq B ≥ tq K B ≥ K t K_B\geq K_t K B ≥ K t (26) . The cases p = 0 p=0 p = 0 q = 0 q=0 q = 0
For affiliated units, clip T T T X X X [ 1 / R , R ] [1/R,R] [ 1/ R , R ] 1 / R < t < R 1/R<t<R 1/ R < t < R T T T t t t T , X T,X T , X 7.2 7.2 7.2 (7.3) show that the resulting positive contractions converge in L 1 L^1 L 1
An affiliated positive mean
Fix a shift-invariant mean m m m ℓ ∞ ( Z ) \ell^\infty(\mathbb{Z}) ℓ ∞ ( Z ) [ − N , N ] [-N,N] [ − N , N ] Y n ∈ M Y_n\in\mathcal{M} Y n ∈ M M n Y n ∈ M \mathfrak{M}_nY_n\in\mathcal{M} M n Y n ∈ M
φ ( M n Y n ) = m ( n ↦ φ ( Y n ) ) , φ ∈ M ∗ . \varphi(\mathfrak{M}_nY_n)=m\left(n\mapsto\varphi(Y_n)\right),\qquad\varphi\in\mathcal{M}_*. φ ( M n Y n ) = m ( n ↦ φ ( Y n ) ) , φ ∈ M ∗ . This mean is positive and unital, commutes with fixed bounded left and right multiplications, and is shift invariant. It also satisfies
∥ M n Y n ∥ 1 ≤ sup n ∥ Y n ∥ 1 . (27) \|\mathfrak{M}_nY_n\|_1\leq\sup_n\|Y_n\|_1. \tag*{(27)} ∥ M n Y n ∥ 1 ≤ n sup ∥ Y n ∥ 1 . (27) Indeed, pair with an arbitrary element of the unit ball of M \mathcal{M} M L 1 L^1 L 1
For finitely many bounded positive invertible inputs and a weighted average, the equation below defines the induced operator mean associated with the binary harmonic mean [56 ] . We prove the affiliated version needed here, under tightness of the inputs and their inverses and with the fixed shift-invariant mean m m m
Proposition 7.4 (Affiliated positive mean). Let X n X_n X n n ∈ Z n\in\mathbb{Z} n ∈ Z { X n } \{X_n\} { X n } { X n − 1 } \{X_n^{-1}\} { X n − 1 } P P P
M n F ( P , X n ) = 1 2 1. (28) \mathfrak{M}_nF(P,X_n)=\tfrac{1}{2}1. \tag*{(28)} M n F ( P , X n ) = 2 1 1. (28) The construction is covariant under congruence by affiliated invertible operators, including affiliated invertible maps between corners with their restricted traces.
Proof. We divide the argument into bounded construction, uniform tails, comparison, and passage to the limit. For an integer R ≥ 1 R \ge1 R ≥ 1 X n , R X_{n,R} X n , R X n X_n X n [ R − 1 , R ] [R^{-1},R] [ R − 1 , R ]
H ( P , X ) = ( P − 1 + X − 1 ) − 1 , Φ R ( P ) = 2 M n H ( P , X n , R ) . H(P,X)=(P^{-1}+X^{-1})^{-1}, \qquad\Phi_R(P)=2\mathcal{M}_nH(P,X_{n,R}). H ( P , X ) = ( P − 1 + X − 1 ) − 1 , Φ R ( P ) = 2 M n H ( P , X n , R ) . Inversion reverses positive order, so Φ R \Phi_R Φ R [ R − 1 1 , R 1 ] [R^{-1}1,R1] [ R − 1 1 , R 1 ] P ( 0 ) = R 1 P^{(0)}=R1 P ( 0 ) = R 1 P ( j + 1 ) = Φ R ( P ( j ) ) P^{(j+1)}=\Phi_R(P^{(j)}) P ( j + 1 ) = Φ R ( P ( j ) ) P R P_R P R L 1 L^1 L 1
This convergence passes through Φ R \Phi_R Φ R n n n
H ( P , X ) = X − X ( P + X ) − 1 X , H(P,X)=X-X(P+X)^{-1}X, H ( P , X ) = X − X ( P + X ) − 1 X , and the resolvent identity gives, on the fixed interval in question,
∥ H ( P , X ) − H ( Q , X ) ∥ 1 ≤ C R ∥ P − Q ∥ 1 \|H(P,X)-H(Q,X)\|_1\le C_R\|P-Q\|_1 ∥ H ( P , X ) − H ( Q , X ) ∥ 1 ≤ C R ∥ P − Q ∥ 1 with a constant independent of X = X n , R X=X_{n,R} X = X n , R (27) therefore shows that P R = Φ R ( P R ) P_R=\Phi_R(P_R) P R = Φ R ( P R )
H ( P , X ) = P 1 / 2 ( 1 − F ( P , X ) ) P 1 / 2 (29) H(P,X)=P^{1/2}(1-F(P,X))P^{1/2} \tag*{(29)} H ( P , X ) = P 1/2 ( 1 − F ( P , X )) P 1/2 (29) and bounded invertibility of P R P_R P R
M n F ( P R , X n , R ) = 1 2 1. (30) \mathcal{M}_nF(P_R,X_{n,R})=\tfrac{1}{2}1. \tag*{(30)} M n F ( P R , X n , R ) = 2 1 1. (30) The clipped families { X n , R : n ∈ Z , R ≥ 1 } \{X_{n,R}:n\in\mathbb{Z},R\ge1\} { X n , R : n ∈ Z , R ≥ 1 } K > 1 K>1 K > 1
sup n , R ∥ 1 − F ( t 1 , X n , R ) ∥ 1 ⟶ 0 ( t ⟶ ∞ ) , \sup_{n,R}\|1-F(t1,X_{n,R})\|_1\longrightarrow0 \qquad(t\longrightarrow\infty), n , R sup ∥1 − F ( t 1 , X n , R ) ∥ 1 ⟶ 0 ( t ⟶ ∞ ) , sup n , R ∥ F ( s 1 , X n , R ) ∥ 1 ⟶ 0 ( s ⟶ 0 ) . \sup_{n,R}\|F(s1,X_{n,R})\|_1\longrightarrow0 \qquad(s\longrightarrow0). n , R sup ∥ F ( s 1 , X n , R ) ∥ 1 ⟶ 0 ( s ⟶ 0 ) . For example, cutting X n , R X_{n,R} X n , R K K K
K t τ ( 1 ) + sup n , R τ ( 1 ( K , ∞ ) ( X n , R ) ) . \frac{K}{t}\tau(1)+\sup_{n,R}\tau(1_{(K,\infty)}(X_{n,R})). t K τ ( 1 ) + n , R sup τ ( 1 ( K , ∞ ) ( X n , R )) . For the second, cut X n , R − 1 X_{n,R}^{-1} X n , R − 1 K K K s K τ ( 1 ) sK\tau(1) sK τ ( 1 ) (7.3) to p = 1 ( t , ∞ ) ( P R ) p=1_{(t,\infty)}(P_R) p = 1 ( t , ∞ ) ( P R ) (30) . It gives
1 2 τ ( p ) ≥ τ ( p ) − sup n ∥ 1 − F ( t 1 , X n , R ) ∥ 1 . \tfrac{1}{2}\tau(p)\ge\tau(p)-\sup_n\|1-F(t1,X_{n,R})\|_1. 2 1 τ ( p ) ≥ τ ( p ) − n sup ∥1 − F ( t 1 , X n , R ) ∥ 1 . For q = 1 ( 0 , s ) ( P R ) q=1_{(0,s)}(P_R) q = 1 ( 0 , s ) ( P R )
1 2 τ ( q ) ≤ sup n ∥ F ( s 1 , X n , R ) ∥ 1 . \tfrac{1}{2}\tau(q)\le\sup_n\|F(s1,X_{n,R})\|_1. 2 1 τ ( q ) ≤ n sup ∥ F ( s 1 , X n , R ) ∥ 1 . Thus both { P R } \{P_R\} { P R } { P R − 1 } \{P_R^{-1}\} { P R − 1 } R R R
We next remove clipping from the equation. Put
q n , R = 1 ( 0 , R − 1 ) ∪ ( R , ∞ ) ( X n ) . q_{n,R}=1_{(0,R^{-1})\cup(R,\infty)}(X_n). q n , R = 1 ( 0 , R − 1 ) ∪ ( R , ∞ ) ( X n ) . Two-sided tightness implies sup n τ ( q n , R ) → 0 \sup_n\tau(q_{n,R})\to0 sup n τ ( q n , R ) → 0
F ( P R , X n ) − F ( P R , X n , R ) = P R 1 / 2 ( P R + X n ) − 1 ⋅ ( X n , R − X n ) ( P R + X n , R ) − 1 P R 1 / 2 . F(P_R,X_n)-F(P_R,X_{n,R})=P_R^{1/2}(P_R+X_n)^{-1}
\cdot(X_{n,R}-X_n)(P_R+X_{n,R})^{-1}P_R^{1/2}. F ( P R , X n ) − F ( P R , X n , R ) = P R 1/2 ( P R + X n ) − 1 ⋅ ( X n , R − X n ) ( P R + X n , R ) − 1 P R 1/2 . The middle factor is supported on q n , R q_{n,R} q n , R
sup n ∥ F ( P R , X n ) − F ( P R , X n , R ) ∥ 1 ≤ 2 sup n τ ( q n , R ) ⟶ 0. \sup_n \left\|F(P_R,X_n)-F(P_R,X_{n,R})\right\|_1 \le2\sup_n\tau(q_{n,R})\longrightarrow0. n sup ∥ F ( P R , X n ) − F ( P R , X n , R ) ∥ 1 ≤ 2 n sup τ ( q n , R ) ⟶ 0. It follows that
E R : = M n F ( P R , X n ) − 1 2 1 E_R:=\mathfrak{M}_nF(P_R,X_n)-\frac{1}{2}1 E R := M n F ( P R , X n ) − 2 1 1 satisfies ∥ E R ∥ 1 ⟶ 0 \|E_R\|_1\longrightarrow0 ∥ E R ∥ 1 ⟶ 0
We record precisely the congruence rule. If C C C P ′ = C P C ∗ P'=CPC^* P ′ = C P C ∗ X ′ = C X C ∗ X'=CXC^* X ′ = C X C ∗
F ( P ′ , X ′ ) = U F ( P , X ) U ∗ , U = ( P ′ ) − 1 / 2 C P 1 / 2 . (31) F(P',X')=UF(P,X)U^*,\qquad U=(P')^{-1/2}CP^{1/2}. \tag*{(31)} F ( P ′ , X ′ ) = U F ( P , X ) U ∗ , U = ( P ′ ) − 1/2 C P 1/2 . (31) The affiliated algebra identities U U ∗ = U ∗ U = 1 UU^*=U^*U=1 U U ∗ = U ∗ U = 1 U U U C ( P + X ) C ∗ C(P+X)C^* C ( P + X ) C ∗ U U U U U U P , C P,C P , C X X X L 1 L^1 L 1
We now prove that P R P_R P R R , S R,S R , S
C = P R − 1 / 2 , T = C P S C , X n ′ = C X n C . C=P_R^{-1/2},\qquad T=CP_SC,\qquad X'_n=CX_nC. C = P R − 1/2 , T = C P S C , X n ′ = C X n C . The equations with errors E R , E S E_R,E_S E R , E S
M n F ( 1 , X n ′ ) = 1 2 1 + E R , M n F ( T , X n ′ ) = 1 2 1 + U E S U ∗ \mathfrak{M}_nF(1,X'_n)=\frac{1}{2}1+E_R,\qquad\mathfrak{M}_nF(T,X'_n)=\frac{1}{2}1+UE_SU^* M n F ( 1 , X n ′ ) = 2 1 1 + E R , M n F ( T , X n ′ ) = 2 1 1 + U E S U ∗ for a unitary U U U n n n 1 1 1 [7 ] , the families X n ′ X'_n X n ′ ( X n ′ ) − 1 (X'_n)^{-1} ( X n ′ ) − 1 R R R n n n
Fix d , ε > 0 d,\varepsilon>0 d , ε > 0 p = 1 ( 1 + d , ∞ ) ( T ) p=1_{(1+d,\infty)}(T) p = 1 ( 1 + d , ∞ ) ( T ) τ ( p ) ≥ ε \tau(p)\ge\varepsilon τ ( p ) ≥ ε
∥ E R ∥ 1 + ∥ E S ∥ 1 ≥ τ ( p M n δ d ( X n ′ ) ) , δ d ( x ) = d x ( 1 + d + x ) ( 1 + x ) . (32) \|E_R\|_1+\|E_S\|_1\ge\tau\bigl(p\mathfrak{M}_n\delta_d(X'_n)\bigr),\qquad\delta_d(x)=\frac{dx}{(1+d+x)(1+x)}. \tag*{(32)} ∥ E R ∥ 1 + ∥ E S ∥ 1 ≥ τ ( p M n δ d ( X n ′ ) ) , δ d ( x ) = ( 1 + d + x ) ( 1 + x ) d x . (32) Choose 0 < a < b < ∞ 0<a<b<\infty 0 < a < b < ∞ R , n R,n R , n
τ ( 1 − 1 [ a , b ] ( X n ′ ) ) < ε / 2. \tau\bigl(1-1_{[a,b]}(X'_n)\bigr)<\varepsilon/2. τ ( 1 − 1 [ a , b ] ( X n ′ ) ) < ε /2. On [ a , b ] [a,b] [ a , b ] δ d \delta_d δ d c c c q n = 1 [ a , b ] ( X n ′ ) q_n=1_{[a,b]}(X'_n) q n = 1 [ a , b ] ( X n ′ )
τ ( p δ d ( X n ′ ) ) ≥ c τ ( p q n ) ≥ c ( τ ( p ) − τ ( 1 − q n ) ) ≥ c ε / 2. \tau\bigl(p\delta_d(X'_n)\bigr)\ge c\tau(pq_n)\ge c\bigl(\tau(p)-\tau(1-q_n)\bigr)\ge c\varepsilon/2. τ ( p δ d ( X n ′ ) ) ≥ c τ ( p q n ) ≥ c ( τ ( p ) − τ ( 1 − q n ) ) ≥ c ε /2. These inequalities do not require p p p q n q_n q n n n n R , S R,S R , S
For the low projection p = 1 ( 0 , 1 − d ) ( T ) p=1_{(0,1-d)}(T) p = 1 ( 0 , 1 − d ) ( T ) 0 < d < 1 0<d<1 0 < d < 1
1 1 + x − 1 − d 1 − d + x = d x ( 1 + x ) ( 1 − d + x ) . \frac{1}{1+x}-\frac{1-d}{1-d+x}=\frac{dx}{(1+x)(1-d+x)}. 1 + x 1 − 1 − d + x 1 − d = ( 1 + x ) ( 1 − d + x ) d x . The identical argument bounds its trace away from zero whenever τ ( p ) ≥ ε \tau(p)\ge\varepsilon τ ( p ) ≥ ε T − 1 ⟶ 0 T-1\longrightarrow0 T − 1 ⟶ 0 R , S → ∞ R,S\to\infty R , S → ∞ T − 1 − 1 ⟶ 0 T^{-1}-1\longrightarrow0 T − 1 − 1 ⟶ 0
P S − P R = P R 1 / 2 ( T − 1 ) P R 1 / 2 , P S − 1 − P R − 1 = P R − 1 / 2 ( T − 1 − 1 ) P R − 1 / 2 , P_S-P_R=P_R^{1/2}(T-1)P_R^{1/2},\qquad P_S^{-1}-P_R^{-1}=P_R^{-1/2}(T^{-1}-1)P_R^{-1/2}, P S − P R = P R 1/2 ( T − 1 ) P R 1/2 , P S − 1 − P R − 1 = P R − 1/2 ( T − 1 − 1 ) P R − 1/2 , tightness and Lemma 7.2 show that both ( P R ) (P_R) ( P R ) ( P R − 1 ) (P_R^{-1}) ( P R − 1 ) P , V P,V P , V P V = V P = 1 PV=VP=1 P V = V P = 1 P P P V = P − 1 V=P^{-1} V = P − 1
P R − 1 / 2 X n P R − 1 / 2 − P − 1 / 2 X n P − 1 / 2 ⟶ 0 P_R^{-1/2}X_nP_R^{-1/2}-P^{-1/2}X_nP^{-1/2}\longrightarrow0 P R − 1/2 X n P R − 1/2 − P − 1/2 X n P − 1/2 ⟶ 0 in measure uniformly in n n n (7.3) gives uniform measure convergence of F ( P R , X n ) F(P_R,X_n) F ( P R , X n ) F ( P , X n ) F(P,X_n) F ( P , X n ) L 1 L^1 L 1 (27) and eq:7.10 prove (28) .
Finally, if P , Q P,Q P , Q (32) . Therefore P − 1 / 2 Q P − 1 / 2 = 1 P^{-1/2}QP^{-1/2}=1 P − 1/2 Q P − 1/2 = 1
Proof of the support lemma
Proof of Lemma 7.1. If λ = 0 \lambda=0 λ = 0 α = Ad g D \alpha=\operatorname{Ad}g_D α = Ad g D L L L S S S L L L
T ( ξ ) = g ξ g D − 1 . \mathcal{T}(\xi)=g\xi g_D^{-1}. T ( ξ ) = g ξ g D − 1 . This is well defined because h ( g ) = h ( g D ) h(g)=h(g_D) h ( g ) = h ( g D ) α \alpha α T ( ξ w ) = T ( ξ ) α ( w ) \mathcal{T}(\xi w)=\mathcal{T}(\xi)\alpha(w) T ( ξ w ) = T ( ξ ) α ( w )
We first identify the column kernel with a projection in M r ( N ( S ) ) M_r(\mathcal{N}(S)) M r ( N ( S ))
v = ( 1 + λ λ ∗ ) − 1 / 2 ∈ U ( L ) , a = v λ . v=(1+\lambda\lambda^*)^{-1/2}\in\mathcal{U}(L),\qquad a=v\lambda. v = ( 1 + λ λ ∗ ) − 1/2 ∈ U ( L ) , a = v λ . The row a a a
H = E S ( a ∗ a ) ∈ M r ( N ( S ) ) , e = supp H , H=\mathbb{E}_S(a^*a)\in M_r(\mathcal{N}(S)),\qquad e=\operatorname{supp}H, H = E S ( a ∗ a ) ∈ M r ( N ( S )) , e = supp H , where E S : N ( L ) → N ( S ) \mathbb{E}_S:\mathcal{N}(L)\to\mathcal{N}(S) E S : N ( L ) → N ( S ) P S : ℓ 2 ( L ) → ℓ 2 ( S ) P_S:\ell^2(L)\to\ell^2(S) P S : ℓ 2 ( L ) → ℓ 2 ( S ) E S ( a ) = P S a ∣ ℓ 2 ( S ) \mathbb{E}_S(a)=P_Sa|_{\ell^2(S)} E S ( a ) = P S a ∣ ℓ 2 ( S ) S S S N ( S ) \mathcal{N}(S) N ( S ) ℓ 2 ( S ) \ell^2(S) ℓ 2 ( S ) δ 1 \delta_1 δ 1 [4 ] . Neither normality nor finite index of S S S S S S w w w
τ ( ( a w ) ∗ a w ) = τ ( w ∗ H w ) . \tau((aw)^*aw)=\tau(w^*Hw). τ (( a w ) ∗ a w ) = τ ( w ∗ H w ) . Faithfulness shows that a w = 0 aw=0 a w = 0 e w = 0 ew=0 e w = 0 ∣ w ∣ |w| ∣ w ∣ w w w
ker ( λ : U ( S ) r → U ( L ) ) = ( 1 − e ) U ( S ) r , λ e = λ . (33) \ker(\lambda:\mathcal{U}(S)^r\to\mathcal{U}(L))=(1-e)\mathcal{U}(S)^r,\qquad\lambda e=\lambda. \tag*{(33)} ker ( λ : U ( S ) r → U ( L )) = ( 1 − e ) U ( S ) r , λ e = λ . (33) Here λ e = λ \lambda e=\lambda λ e = λ a ( 1 − e ) = 0 a(1-e)=0 a ( 1 − e ) = 0 e ≠ 0 e\ne0 e = 0 g n λ g D − n g^n\lambda g_D^{-n} g n λ g D − n
w j = e w j , { λ w j } is tight } ⟹ { w j } is tight . \begin{equation}
\left.
\begin{array}{c}
w_j=ew_j,\\
\{\lambda w_j\}\text{ is tight}
\end{array}\right\}
\quad\Longrightarrow\quad
\{w_j\}\text{ is tight}.
\end{equation} w j = e w j , { λ w j } is tight } ⟹ { w j } is tight . The matrices w j w_j w j p = 1 ( R , ∞ ) ( w w ∗ ) ≤ e p=1_{(R,\infty)}(ww^*)\le e p = 1 ( R , ∞ ) ( w w ∗ ) ≤ e τ r ( p ) ≥ ε \tau_r(p)\ge\varepsilon τ r ( p ) ≥ ε supp H = e \operatorname{supp}H=e supp H = e δ > 0 \delta>0 δ > 0
τ r ( e − 1 [ δ , ∞ ) ( H ) ) < ε / 2. \tau_r\left(e-1_{[\delta,\infty)}(H)\right)<\varepsilon/2. τ r ( e − 1 [ δ , ∞ ) ( H ) ) < ε /2. Then, writing q = 1 [ δ , ∞ ) ( H ) q=1_{[\delta,\infty)}(H) q = 1 [ δ , ∞ ) ( H )
τ ( a p a ∗ ) = τ r ( p H ) ≥ δ τ r ( p q ) ≥ δ ε / 2 = : c > 0. \tau(apa^*)=\tau_r(pH)\ge\delta\tau_r(pq)\ge\delta\varepsilon/2=:c>0. τ ( a p a ∗ ) = τ r ( p H ) ≥ δ τ r ( pq ) ≥ δ ε /2 =: c > 0. Also 0 ≤ a p a ∗ ≤ ∥ a ∥ 2 1 0\le apa^*\le\lVert a\rVert^2 1 0 ≤ a p a ∗ ≤ ∥ a ∥ 2 1 a p a ∗ apa^* a p a ∗ c / ( 2 ∥ a ∥ 2 ) c/(2\lVert a\rVert^2) c / ( 2 ∥ a ∥ 2 ) c / 2 c/2 c /2 τ ( 1 ) = 1 \tau(1)=1 τ ( 1 ) = 1 N ( L ) \mathcal{N}(L) N ( L )
a w w ∗ a ∗ ≥ R a p a ∗ . aww^*a^*\ge Rapa^*. a w w ∗ a ∗ ≥ R a p a ∗ . Monotonicity of spectral distribution under positive order therefore forces the same positive amount of spectral mass above R c / 2 Rc/2 R c /2 a w w ∗ a ∗ aww^*a^* a w w ∗ a ∗ a w aw a w R → ∞ R\to\infty R → ∞ a w = v ( λ w ) aw=v(\lambda w) a w = v ( λ w ) λ w \lambda w λ w a w aw a w (7.14) .
Define the full invertible cocycle C n C_n C n n ∈ Z n\in\mathbb{Z} n ∈ Z
C 0 = 1 , C 1 = B , C n + m = C n α n ( C m ) . C_0=1,\qquad C_1=B,\qquad C_{n+m}=C_n\alpha^n(C_m). C 0 = 1 , C 1 = B , C n + m = C n α n ( C m ) . Iterating (7.1) gives
T n ( λ ) = g n λ g D − n = λ C n . \mathcal{T}^n(\lambda)=g^n\lambda g^{-n}_D=\lambda C_n. T n ( λ ) = g n λ g D − n = λ C n . Semilinearity and (33) imply
C n α n ( ( 1 − e ) U ( S ) r ) = ( 1 − e ) U ( S ) r . C_n\alpha^n((1-e)U(S)^r)=(1-e)U(S)^r. C n α n (( 1 − e ) U ( S ) r ) = ( 1 − e ) U ( S ) r . Hence the induced maps on the quotient columns have representatives
b n = e C n α n ( e ) : α n ( e ) U ( S ) r ⟶ e U ( S ) r . (34) b_n=eC_n\alpha^n(e):\alpha^n(e)U(S)^r\longrightarrow eU(S)^r. \tag*{(34)} b n = e C n α n ( e ) : α n ( e ) U ( S ) r ⟶ e U ( S ) r . (34) Their inverses are
d n = α n ( e ) C n − 1 e = α n ( b − n ) , b n d n = e , d n b n = α n ( e ) . d_n=\alpha^n(e)C_n^{-1}e=\alpha^n(b_{-n}),\qquad b_nd_n=e,\qquad d_nb_n=\alpha^n(e). d n = α n ( e ) C n − 1 e = α n ( b − n ) , b n d n = e , d n b n = α n ( e ) . Compression does not spoil the cocycle law. Indeed the discarded summand α n ( 1 − e ) \alpha^n(1-e) α n ( 1 − e ) C n C_n C n e e e
b n + m = b n α n ( b m ) . (35) b_{n+m}=b_n\alpha^n(b_m). \tag*{(35)} b n + m = b n α n ( b m ) . (35) Moreover,
λ b n = λ C n α n ( e ) = T n ( λ ) . (36) \lambda b_n=\lambda C_n\alpha^n(e)=\mathcal{T}^n(\lambda). \tag*{(36)} λ b n = λ C n α n ( e ) = T n ( λ ) . (36) The spectral distributions of the rows on the right of (36) are independent of n n n α n ( λ ∗ λ ) \alpha^n(\lambda^*\lambda) α n ( λ ∗ λ ) e b n = b n eb_n=b_n e b n = b n (7.14) shows that { b n } \{b_n\} { b n } α \alpha α d n d_n d n { d n } \{d_n\} { d n } e M r ( U ( S ) ) e eM_r(U(S))e e M r ( U ( S )) e
X n = b n b n ∗ , X n − 1 = d n ∗ d n . X_n=b_nb_n^*,\qquad X_n^{-1}=d_n^*d_n. X n = b n b n ∗ , X n − 1 = d n ∗ d n . Both families are tight, by the product estimate. Apply Proposition 7.4 in this corner to obtain the unique positive affiliated unit P P P
M n F ( P , X n ) = 1 2 e . \mathfrak{M}_nF(P,X_n)=\tfrac{1}{2}e. M n F ( P , X n ) = 2 1 e . Equation (35) gives
X n + 1 = b 1 α ( X n ) b 1 ∗ X_{n+1}=b_1\alpha(X_n)b_1^* X n + 1 = b 1 α ( X n ) b 1 ∗ The automorphism α \alpha α α ( e ) \alpha(e) α ( e ) b 1 : α ( e ) → e b_1:\alpha(e)\to e b 1 : α ( e ) → e b 1 α ( P ) b 1 ∗ b_1\alpha(P)b_1^* b 1 α ( P ) b 1 ∗ X n + 1 X_{n+1} X n + 1
P = b 1 α ( P ) b 1 ∗ . (37) P=b_1\alpha(P)b_1^*. \tag*{(37)} P = b 1 α ( P ) b 1 ∗ . (37) Set
Z = λ P 1 / 2 , u = P − 1 / 2 b 1 α ( P ) 1 / 2 . Z=\lambda P^{1/2},\qquad u=P^{-1/2}b_1\alpha(P)^{1/2}. Z = λ P 1/2 , u = P − 1/2 b 1 α ( P ) 1/2 . Inverses of P P P e e e (37) gives u u ∗ = e uu^*=e u u ∗ = e u ∗ u = α ( e ) u^*u=\alpha(e) u ∗ u = α ( e ) u u u N ( S ) \mathcal{N}(S) N ( S ) (36) and (37) imply
g Z g D − 1 = Z u , g Z Z ∗ g D − 1 = Z u u ∗ Z ∗ = Z Z ∗ . gZg_D^{-1}=Zu,\qquad gZZ^*g_D^{-1}=Zuu^*Z^*=ZZ^*. g Z g D − 1 = Z u , g Z Z ∗ g D − 1 = Z u u ∗ Z ∗ = Z Z ∗ . Every conjugation-invariant positive affiliated operator in U ( L ) \mathcal{U}(L) U ( L ) g g g l ∈ L l\in L l ∈ L g g g g m g^m g m g m g^m g m l l l N ( L ) \mathcal{N}(L) N ( L )
Applying this to Z Z ∗ ZZ^* Z Z ∗ Z Z ∗ = c 1 ZZ^*=c1 Z Z ∗ = c 1 c ≥ 0 c\geq0 c ≥ 0 Z j Z_j Z j Z j Z j ∗ ≤ c 1 Z_jZ_j^*\leq c1 Z j Z j ∗ ≤ c 1 H ⊆ ℓ 2 ( L ) \mathcal{H}\subseteq\ell^2(L) H ⊆ ℓ 2 ( L ) S S S Z 1 , … , Z r Z_1,\ldots,Z_r Z 1 , … , Z r
ℓ 2 ( S ) r ⟶ ℓ 2 ( L ) , ( ξ 1 , … , ξ r ) ⟼ ∑ j = 1 r Z j ξ j \ell^2(S)^r\longrightarrow\ell^2(L),\qquad
(\xi_1,\ldots,\xi_r)\longmapsto\sum_{j=1}^{r}Z_j\xi_j ℓ 2 ( S ) r ⟶ ℓ 2 ( L ) , ( ξ 1 , … , ξ r ) ⟼ j = 1 ∑ r Z j ξ j is bounded and right S S S H \mathcal{H} H S S S H \mathcal{H} H r r r [42 ] . The unitary T : ξ ↦ g ξ g D − 1 T:\xi\mapsto g\xi g_D^{-1} T : ξ ↦ g ξ g D − 1 ℓ 2 ( L ) \ell^2(L) ℓ 2 ( L ) H \mathcal{H} H T ( Z ) = Z u T(Z)=Zu T ( Z ) = Z u T ( Z ) u ∗ = Z T(Z)u^*=Z T ( Z ) u ∗ = Z
Decompose ℓ 2 ( L ) \ell^2(L) ℓ 2 ( L ) x S xS x S S S S Π \Pi Π H \mathcal{H} H x S xS x S
t x S = ⟨ Π δ x , δ x ⟩ ≥ 0. t_{xS}=\langle\Pi\delta_x,\delta_x\rangle\geq0. t x S = ⟨ Π δ x , δ x ⟩ ≥ 0. It is independent of the representative x x x
∑ x S ∈ L / S t x S = dim S H ≤ r . \sum_{xS\in L/S}t_{xS}=\dim_S\mathcal{H}\leq r. x S ∈ L / S ∑ t x S = dim S H ≤ r . The unitary T T T
x S ⟼ g x g D − 1 S . xS\longmapsto gxg_D^{-1}S. x S ⟼ g x g D − 1 S . It commutes with Π \Pi Π t x S = 0 t_{xS}=0 t x S = 0 S S S Π \Pi Π Π δ x = 0 \Pi\delta_x=0 Π δ x = 0 s ∈ S s\in S s ∈ S H \mathcal{H} H Z Z Z
A coset is periodic precisely when, for some m ≠ 0 m \ne0 m = 0
g m x g D − m S = x S . g^{m}xg_{D}^{-m}S=xS. g m x g D − m S = x S . This implies x − 1 g m x ∈ D x^{-1}g^{m}x \in D x − 1 g m x ∈ D x − 1 g m x ∈ D x^{-1}g^{m}x \in D x − 1 g m x ∈ D x − 1 g m x g D − m x^{-1}g^{m}xg_{D}^{-m} x − 1 g m x g D − m D D D S S S P \mathcal{P} P
Finally, P \mathcal{P} P D D D x S ∈ P xS \in\mathcal{P} x S ∈ P g g g x D xD x D g + , g − g^{+},g^{-} g + , g − x D xD x D x D xD x D x D xD x D x ∈ L x \in L x ∈ L
x D ∩ L = x S , xD \cap L=xS, x D ∩ L = x S , and distinct x S xS x S P \mathcal{P} P x D xD x D
Choose one representative x ∈ L x \in L x ∈ L P \mathcal{P} P
Z j = ∑ x S ∈ P x E S ( x − 1 Z j ) , E S ( x − 1 Z j ) ∈ N ( S ) . Z_j=\sum_{xS\in\mathcal{P}}x\,E_S(x^{-1}Z_j),\qquad E_S(x^{-1}Z_j)\in\mathcal{N}(S). Z j = x S ∈ P ∑ x E S ( x − 1 Z j ) , E S ( x − 1 Z j ) ∈ N ( S ) . This is an equality first in ℓ 2 ( L ) \ell^2(L) ℓ 2 ( L ) N ( L ) \mathcal{N}(L) N ( L ) P − 1 / 2 P^{-1/2} P − 1/2 λ \lambda λ λ e = λ \lambda e=\lambda λ e = λ S S S
Expansion gates
We show that a sufficiently long carried arc expands in one direction, and that enough expansion determines a unique continuation in that direction. These path lemmas will be applied to the ordinary rows and, later, to relative rows. The relative statement is conditional on the full package below; passing to relative rows does not automatically preserve that package. We retain the ambient torsion-free hyperbolic group, its height homomorphism, and the cyclic noncommensuration property established above. In particular, a nonzero-height element cannot commensurate a nontrivial height-zero cyclic subgroup.
Carried arcs and bounded comparisons
For the ordinary application, sample successive coarse Magnus edges of T 0 T_0 T 0 H i H_i H i R i R_i R i
Definition 8.1 (The expansion package). Consider oriented monotone paths in a tree of groups with height. Sample at successive occurrences of one edge type, and number these occurrences by integers. At position i i i W i W_i W i s s s H i = Stab ( W i ) H_i=\operatorname{Stab}(W_i) H i = Stab ( W i ) nonparabolic if it is nontrivial and is not contained in any one of these actual peripheral groups. The required package consists of the following properties. (E1) The total tree of ordinary or relative Cayley spaces is hyperbolic, and all adjacent edge maps are uniformly quasi-isometric embeddings.
(E2) Nonparabolically stabilized windows of each fixed shape have finitely many orbits. The same holds for their nonparabolically stabilized extensions, including configurations with a specified common subwindow. Their stabilizers, with intrinsic relative metrics, embed quasi-isometrically in every row of the window; constants are uniform for each bounded shape.
(E3) A stabilizer of a window containing a short window is a free product of representatives of its nontrivial peripheral intersections and a free group, with a finite relative decomposition. The peripheral systems restrict exactly under further window restrictions. Every window group under consideration has a loxodromic element in the relevant short-window free-product trees.
(E4) Two distinct continuations through the same short window, with their terminal positions at the same height, have stabilizers intersecting in an at most cyclic group.
For H i H_i H i R i R_i R i R i R_i R i
A trajectory is one of these monotone paths through the row tree; it may be finite, one-sided infinite, or bilateral. The arcs carried along it lie in the corresponding trees R i R_i R i R i R_i R i core means its minimal invariant subtree; its quotient will be called the core graph . These are different objects: the subtree is generally infinite, while the core graphs below have finitely many edges. An edge of R i R_i R i regular edge , to emphasize its trivial stabilizer. An arc is carried by a window if it lies on a bi-infinite line in that window group’s core. For an infinite path, carrying means carrying by every relevant finite subwindow.
Lemma 8.2 (Bounded comparisons). Assume the expansion package. For a fixed bounded pair of positions i , j i,j i , j K K K W i ∪ W j W_i \cup W_j W i ∪ W j K K K K K K R i R_i R i R j R_j R j
Proof. The K K K K K K
For each representative vertex choose an image fixed by its stabilizer in the other tree; such a vertex exists by exact ellipticity. Extend these choices equivariantly and then over edges. Finite quotient gives bounded choices, coarse inverses, and the asserted quasi-isometries. For a fixed finite diagram, first compare the maps and their composites on the core of the common window spanned by that diagram. They are equivariant for its group and agree within bounded distance on a finite set of orbit representatives. Property (E2) makes this bound uniform for the bounded shape of the diagram. A further window has its core inside this common core, so restriction inherits the same bound; no coboundedness constant depending on that further window is introduced. This proves compatibility and preservation of the boundary points of further window cores.
We use the following precise convention for arcs. Join the images of the endpoints by a geodesic and delete a fixed bounded segment at each end. The remaining arc lies in every transferred line containing the original arc. The trimming constant follows from stability of quasi-geodesics in a tree. Under immediate inverse comparison, the transferred arc is contained in a bounded enlargement of the original arc; in particular its length is at most the original length plus a uniform additive constant.
The corridor estimate and phase uniqueness
Lemma 8.3 (Flaring). Under the expansion package there are integers b b b R > 0 R > 0 R > 0 R R R [ − b , b ] [-b,b] [ − b , b ] 0 → − b 0 \to-b 0 → − b 0 → b 0 \to b 0 → b
Proof. We establish the corridor estimate with its dependence on the window length made explicit. Fix m m m [ − m , m ] [-m,m] [ − m , m ] m m m m m m
Let l j l_j l j j j j l 0 l_0 l 0 l j l_j l j m m m
For d ≤ m d \le m d ≤ m − d -d − d d d d [12 ] . Here the Rips parameter and the linear isoperimetric constant are fixed. We explain why its area bounds the sum of the intervening fiber widths. Extend projection to the underlying tree across the edges of the triangulation by bounded paths and across triangles by tripod fillings. A transverse cut through an intervening tree edge has two boundary crossings, one on each section. A component of its preimage joins those crossings: preimage tracks pair boundary crossings, and these are the only two. A small perturbation arranges transversality without altering the boundary projection.
Label consecutive track crossings by points of the relevant actual fiber within bounded total-space distance of the triangle vertices. Such points exist because a bounded path whose projected endpoints lie on opposite sides crosses the fiber. Two resulting nearby fiber points have uniformly bounded intrinsic distance. Indeed a bounded total-space path between them has a bounded projected edge word; successively remove its backtracks, using the uniform quasi-isometric edge embeddings. This bounds the intrinsic fiber displacement by a constant depending only on the fixed Rips parameter. Each triangle meets only a uniformly bounded number of sampled cuts. Summing over tracks therefore costs only a fixed multiple of the number of triangles. The horizontal boundary lengths and the bounded section lengths give
∑ − d < j < d l j ≤ C ( l − d + l d ) + C m , 1 ≤ d ≤ m , \sum_{-d<j<d} l_j \le C(l_{-d}+l_d)+C_m,\qquad1\le d\le m, − d < j < d ∑ l j ≤ C ( l − d + l d ) + C m , 1 ≤ d ≤ m , where C C C m m m
Put P d = ∑ ∣ j ∣ ≤ d l j P_d=\sum_{|j|\le d}l_j P d = ∑ ∣ j ∣ ≤ d l j (8.1) implies
P d ≥ ( 1 + C − 1 ) P d − 1 − C m / C . P_d\ge(1+C^{-1})P_{d-1}-C_m/C. P d ≥ ( 1 + C − 1 ) P d − 1 − C m / C . With m m m m m m l 0 l_0 l 0 m m m m m m m = b m=b m = b l 0 l_0 l 0 R R R
Choose R R R R R R 2 R 2R 2 R expanding event of sign ε \varepsilon ε ε b \varepsilon b ε b 3 ( 2 R ) 3(2R) 3 ( 2 R ) b b b 2 R 2R 2 R
The next lemma isolates the cyclic obstruction used in phase uniqueness. It also supplies the infinite-intersection assertion needed later: a rational line surviving all finite windows will have a common nonzero translation. The finite datum in the proof is an actual pair of continuation windows with a marked edge, not an isomorphism type of free groups.
Lemma 8.4 (Cyclic indices along a trajectory ). Assume the expansion package. Fix a bounded sampling step, and let W j W_j W j e j e_j e j
H j = Stab ( W j ) , K j = Stab ( Hull ( W j ∪ W j + 1 ) ) . H_j=\operatorname{Stab}(W_j),\qquad K_j=\operatorname{Stab}(\operatorname{Hull}(W_j\cup W_{j+1})). H j = Stab ( W j ) , K j = Stab ( Hull ( W j ∪ W j + 1 )) . Let C = ⟨ c 0 ⟩ ≤ L C=\langle c_0\rangle\le L C = ⟨ c 0 ⟩ ≤ L G G G C ∩ K j C\cap K_j C ∩ K j
C ∩ H j = ⟨ c 0 q j ⟩ , C ∩ K j = ⟨ c 0 m j ⟩ , q j , m j > 0. C\cap H_j=\langle c_0^{q_j}\rangle,\qquad C\cap K_j=\langle c_0^{m_j}\rangle,\qquad q_j,m_j>0. C ∩ H j = ⟨ c 0 q j ⟩ , C ∩ K j = ⟨ c 0 m j ⟩ , q j , m j > 0. There are uniform constants B , E B,E B , E m j / q j m_j/q_j m j / q j m j / q j + 1 m_j/q_{j+1} m j / q j + 1 B B B E E E B − E B^{-E} B − E B E B^E B E C C C K j K_j K j
Proof. Let V j = Hull ( W j ∪ W j + 1 ) V_j=\operatorname{Hull}(W_j\cup W_{j+1}) V j = Hull ( W j ∪ W j + 1 ) r j = c 0 q j r_j=c_0^{q_j} r j = c 0 q j r j r_j r j R j R_j R j K j K_j K j r j r_j r j K j K_j K j r j r_j r j K j K_j K j r j r_j r j [ C ∩ H j : C ∩ K j ] = m j / q j [C\cap H_j:C\cap K_j]=m_j/q_j [ C ∩ H j : C ∩ K j ] = m j / q j R j + 1 R_{j+1} R j + 1
Suppose m j / q j > 1 m_j/q_j>1 m j / q j > 1 r j r_j r j W j W_j W j V j V_j V j V j V_j V j r j V j r_jV_j r j V j W j W_j W j K j K_j K j r j K j r j − 1 r_jK_jr_j^{-1} r j K j r j − 1 C ∩ K j C \cap K_j C ∩ K j G G G C C C
Mark an oriented regular edge on their common axis in R j R_j R j W j + 1 W_{j+1} W j + 1
If one marked-pair type occurred at two different heights, its transporter g ∈ G g \in G g ∈ G C C C G G G g g g C C C h ( g ) = h ( e j ′ ) − h ( e j ) ≠ 0 h(g) = h(e_{j'}) - h(e_j) \ne0 h ( g ) = h ( e j ′ ) − h ( e j ) = 0 Z \mathbb{Z} Z E E E
Except at these finitely many indices, q j = m j = q j + 1 q_j = m_j = q_{j+1} q j = m j = q j + 1 q j q_j q j m j m_j m j M M M 1 ≠ c 0 M ∈ K j 1 \ne c_0^M \in K_j 1 = c 0 M ∈ K j j j j
Corollary 8.5 (Persistence of a rational line). Suppose a bilateral trajectory carries, in R 0 R_0 R 0 c ∈ H 0 c \in H_0 c ∈ H 0 c c c
Proof. For each finite window, its core contains this axis and has finite edge quotient. Repetition of a marked axis edge under powers of c c c c c c C C C c c c Z \mathbb{Z} Z
Lemma 8.6 (Phase uniqueness). There is a uniform depth N N N 2 R 2R 2 R N N N
Proof. The obstruction to late divergence is a long periodic overlap. We pull its periods back to the original event. Adjacent primitive periods can change by a bounded index, but Lemma 8.4 bounds the number of such changes. Exponential expansion will therefore exceed the total loss in this pullback.
Suppose there is a first divergence. At the last available common b b b
We record the finite-state overlap argument. For an oriented edge in the intersection of two cores, record its orbit in each core. There are uniformly finitely many such pairs. If two edges have the same pair of records, the two elements carrying one edge to the other coincide, because an oriented edge has trivial stabilizer. Their common value belongs to the intersection of the two groups. Thus the overlap has uniformly finitely many edge orbits under that intersection. If the intersection is trivial, the overlap has uniformly bounded diameter. A nontrivial cyclic intersection can also be elliptic in a relative tree. In that case project one of its fixed vertices to the invariant overlap; the projection is fixed too. A radial segment from this point contains at most one edge of each cyclic orbit, since the action preserves distance from the fixed point. The finite edge-orbit bound therefore bounds the radius of the overlap. Thus a sufficiently long overlap forces the cyclic intersection to be loxodromic. It then runs along its axis apart from bounded initial and terminal pieces: an off-axis hanging segment cannot contain a repeated oriented-edge state. Its period is uniformly bounded.
Let r j r_j r j j j j
r j a = r j − b a ′ as actual group elements , 1 ≤ a , a ′ ≤ C ′ , r_j^a = r_{j-b}^{a'} \quad\text{as actual group elements}, \qquad1 \leq a,a' \leq C', r j a = r j − b a ′ as actual group elements , 1 ≤ a , a ′ ≤ C ′ , where each side is the least positive power of the corresponding primitive root lying in the comparison group. Exact ellipticity ensures that the common element remains loxodromic on the other side. Repetition bounds the indices by the uniform edge-orbit count. Equivariance of the coarse inverse on the axis implies that if p p p ( a ′ / a ) p − O ( 1 ) (a'/a)p - O(1) ( a ′ / a ) p − O ( 1 )
Every equality above identifies powers as actual elements of G G G C ≤ L C \leq L C ≤ L a ′ / a a'/a a ′ / a B − E B^{-E} B − E B E B^E B E
Let t t t p t p_t p t t t t
p 0 ≥ B − E p t − C 0 B E t p_0 \geq B^{-E}p_t - C_0B^Et p 0 ≥ B − E p t − C 0 B E t for a uniform trimming constant C 0 C_0 C 0 100 t 100^t 10 0 t p t ≥ c 1 100 t − C 1 p_t \geq c_1 100^t - C_1 p t ≥ c 1 10 0 t − C 1 c 1 > 0 c_1 > 0 c 1 > 0 t t t 2 R 2R 2 R 2 R 2R 2 R N N N □ \square □
Lemma 8.7 (Changing the sampling offset). For an ordinary bilateral trajectory, an expanding event of a given sign produces events of that sign at every sufficiently distant row in that direction, not merely on its stride-b b b
Proof. There are only finitely many offsets of the stride-b b b 2 R 2R 2 R
Start with a long arc already expanding on the original subsequence, transport it forward m m m m m m m m m m m m
Actual trajectory clusters
Outside the acylindrical case, we use expansion to construct a G G G G G G T 0 T_0 T 0 G G G k − 1 k-1 k − 1
There are three steps. We first obtain periodic end stabilizers from branching surviving subtrees. We then connect positive and negative ends when they support a common bilateral trajectory. Finally, we show that each connected component of this switch graph is a tree and that its stabilizer captures every sufficiently long path group seen by that component.
Surviving cores and periodic ends
For an infinite monotone path all of whose relevant finite windows have noncyclic stabilizer, intersect their decreasing cores in a fixed horizontal tree. Call the intersection the surviving subtree . We only consider an event or a branch when it belongs to this intersection. Local finiteness implies that any nonempty surviving subtree is a union of full lines: extending a finite surviving arc in each finite-window core and taking subsequences gives extensions in both directions. The same argument preserves a specified surviving arc. For a bilateral path, Lemma 8.2 transfers the endspace of its surviving subtree between rows.
This is a geometric intersection of subtrees, not yet the core of the intersection of their groups. A surviving line need not initially come with a common nontrivial translation. A trajectory or tail is called branching when its surviving subtree has a branch vertex, equivalently three distinct ends. Comparisons make this condition independent of the sampled row. The next approximation lemma supplies the group-theoretic control needed alongside these geometric data.
Lemma 9.1 (Decreasing core graphs). Let K n K_n K n M = ⋂ n K n M=\bigcap_n K_n M = ⋂ n K n M M M K n K_n K n n n n
Proof. Choose a finite subgraph of the core graph of M M M K n K_n K n M M M K n K_n K n K n K_n K n M M M
Lemma 9.2 (Branch states and tail groups). There are finitely many orbits of branching bilateral trajectories and of branching one-sided tails. Every branching bilateral trajectory is periodic under a nonzero-height element; every branching one-sided tail is eventually periodic.
For a monotone ray from an edge e e e z z z
M z , e = Stab ( [ e , z ) ) M_{z,e} = \operatorname{Stab}([e,z)) M z , e = Stab ([ e , z )) be its pointwise stabilizer. This is a finite-rank free group. Its homology injects in sufficiently long finite-prefix homologies, and the finite-path restriction rule also holds on restricting within a later tail group, with the same common-level and orbit conventions. If some M z , e M_{z,e} M z , e Y z Y_z Y z z z z D z = Stab G ( z ) D_z = \operatorname{Stab}_G(z) D z = Stab G ( z )
(i) Y z Y_z Y z D z D_z D z Y z Y_z Y z
(ii) D z ∩ L D_z \cap L D z ∩ L Y z Y_z Y z
(iii) D z D_z D z G G G
Proof. We first produce translators using a finite set of trajectory orbits. The core-graph lemma then gives the tail stabilizers and their restriction maps. These inputs establish transitivity along levels and finally ambient quasiconvexity.
In a noncyclic finite-rank free core, the number of branch-vertex orbits is at most twice the reduced rank. At any finite depth, count pairs consisting of a path and a branch vertex of its core, modulo the horizontal free vertex action. Joint path inertia bounds this count uniformly by 2 ( k − 1 ) 2(k-1) 2 ( k − 1 ) 2 ( k − 1 ) 2(k-1) 2 ( k − 1 ) L L L L L L
Choose a height-one element t t t t − 1 t^{-1} t − 1 t t t t − 1 t^{-1} t − 1 L L L L L L
The tail group M z , e M_{z,e} M z , e M z , f M_{z,f} M z , f f f f
Rays to a common end coalesce; their union Y z Y_z Y z D z D_z D z Y z Y_z Y z Y z Y_z Y z z z z
Compare an arbitrary Y z Y_z Y z f f f M z , f M_{z,f} M z , f M z , f M_{z,f} M z , f M z , f ≤ D z ∩ L M_{z,f} \leq D_z \cap L M z , f ≤ D z ∩ L
A height-zero element preserving z z z z z z D z ∩ L D_z \cap L D z ∩ L Y z Y_z Y z z z z Y z Y_z Y z D z D_z D z Y z Y_z Y z D z D_z D z
Active colors and switches
Color an expanding event on an actual bilateral trajectory by its expanding end and its sign. Lemma 8.6 says that sufficiently large fixed-window data determine this color uniquely. Such colors are called active . A switch trajectory is a bilateral trajectory whose surviving subtree supports events of both signs at some row.
Lemma 9.3 (Periodic switches and noncyclic active tails). Every switch trajectory is periodic. For an active color z z z e e e M z , e M_{z,e} M z , e
Proof. Every vertex of a surviving subtree lies on a full line and admits a centered event in at least one sign, by flaring. If both signs occur at a row, connect their centers in the surviving tree. There are opposite centered events at distance at most one: along the connecting vertex path the available sign changes, or a vertex admits both signs. The fixed-window data of such a nearby pair have finitely many orbits, and phase uniqueness determines both ends from these data.
Suppose first that a switch has both signs at infinitely many rows of a stride-b b b
At a pure row, a surviving line lies in a bounded neighborhood of the orbit of the corresponding tail group. Indeed, its centered events have only finitely many marked types. An element transporting one occurrence of a type to another fixes the marked row and preserves the uniquely determined expanding end. It therefore belongs to the tail group. Choose one representative for each occurring type. Every vertex of the line lies within a fixed distance of a translate of one of those finitely many representatives. This proves the assertion and puts the two endpoints of the line in the limit set of the tail group.
This limit-set conclusion transfers back to an earlier row. Intersect the later tail group with the finite connecting-path group. The result is the earlier tail group, expressed in the later frame. Both groups are finitely generated subgroups of a free row, hence quasiconvex, and their limit-set intersection is exactly the limit set of their intersection. The equivariant comparison transfers this intersection of endspaces back to the earlier row. Apply the argument on both sides of the switch. The endpoints of a central surviving line then lie in the limit set of the bilateral pointwise stabilizer.
If the surviving subtree is not already branching, it is a line; the last conclusion makes it a rational cyclic axis. A nontrivial translation of that axis fixes every window. Every finite-window group is noncyclic, so its core branches somewhere on this axis. Translate the branch point into a fixed translation period. Local finiteness and nesting leave a branch direction present at every depth. This gives a surviving branch, a contradiction. The trajectory is thus branching and periodic by Lemma 9.2.
Now consider an active event on a periodic trajectory. Expansion produces arbitrarily many disjoint events of the same sign. Translate them back by periods to one central row. Repeated marked event types at different horizontal positions give a nontrivial element of M z , e M_{z,e} M z , e
For completeness, a nonperiodic single-sign event requires a separate argument. Far along its expanding stride-b b b M z , e M_{z,e} M z , e
A nearby pair determines a switch, so there are finitely many switch orbits. A single fixed-window event determines its color, giving finitely many color orbits. Finally Lemma 8.7 supplies both signs at all sufficiently far appropriate rows on a periodic switch. Translation by periods brings those signs to any prescribed row.
Form the actual bipartite switch graph C C C
Lemma 9.4 (The switch graph seen by a long window). There is a uniform depth such that the following holds for every centered monotone window W W W Stab ( W ) \operatorname{Stab}(W) Stab ( W ) z z z W ⊂ Y z W \subset Y_z W ⊂ Y z W W W C ( W ) C(W) C ( W ) z z z C C C
Proof. First justify the passage from a finite window to bilateral witnesses. A local state consists of a centered event, or a pair of events with centers at distance at most one, together with a fixed-size path neighborhood large enough for phase uniqueness. There are finitely many normalized states. More generally, fixing an actual regular horizontal marking, there are finitely many carried extensions of any prescribed finite depth: the relevant paths have finitely many orbits and their cores have finite edge quotients, while the marking has trivial stabilizer. Thus the extension sets form a finitely branching inverse system.
If a given state occurs in arbitrarily deep window cores, choose compatible extensions successively. The resulting bilateral path carries its specified event or pair in every finite subwindow. Local finiteness of the horizontal tree preserves the full lines carrying those arcs. States that do not persist in this way have a maximum possible depth. Taking a maximum over the finitely many short states gives a uniform depth beyond which every state in question has a bilateral witness. The witness is initially required to agree only on the fixed-size neighborhood recorded in the state. Lemma 8.6, applied to the witness and the finite window, then forces its expanding ray to agree throughout the corresponding side of W W W W W W
Every vertex in the finite-window core admits a centered event. Assign its expanding color using a bilateral witness. Phase uniqueness makes the assignment unambiguous and equivariant under Stab ( W ) \operatorname{Stab}(W) Stab ( W ) W W W W ⊂ Y z W \subset Y_z W ⊂ Y z W W W W W W W ⊂ Y z W \subset Y_z W ⊂ Y z
Conversely, suppose W ⊂ Y z W \subset Y_z W ⊂ Y z z z z z z z z z z Y z Y_z Y z W W W
It remains to exclude a cycle. At the center, use the convex cores of the tail groups M z , e M_{z,e} M z , e W W W
For sufficiently deep W W W Y z Y_z Y z
Splitting marks and simultaneous free factors
We next pass from individual periodic ends to the full stabilizer of a connected component of switches. To see this group through finite windows, temporarily distinguish incidences at an end according to where their rays cross a chosen level. Widely separated levels cut the component into the local trees already constructed. Moving the levels towards the respective ends joins those pieces. We will show that these joins neither create circuits nor increase the rank of their stabilizers. This is where the actual homology injections, rather than only the rank bound, enter.
Fix a nonisolated component of C \mathcal{C} C A A A S = A ∩ L S=A\cap L S = A ∩ L j , l j,l j , l j j j l l l C j , l \mathcal{C}_{j,l} C j , l S S S j j j l l l
For j j j l l l C ( W ) \mathcal{C}(W) C ( W ) [ j , l ] [j,l] [ j , l ] S S S
K ( W ) = Stab ( W ) . K(W)=\operatorname{Stab}(W). K ( W ) = Stab ( W ) . Indeed the two marks determine the segment W W W M z , e M_{z,e} M z , e
M τ = Stab ( τ ) , M_{\tau}=\operatorname{Stab}(\tau), M τ = Stab ( τ ) , the pointwise stabilizers of the bilateral switch trajectories. The same finite-core approximation used for tails shows that M τ M_{\tau} M τ M τ M_{\tau} M τ τ \tau τ
Lemma 9.5 (Orbit control under coarsening). The components of each C j , l \mathcal{C}_{j,l} C j , l S S S j j j l l l
Proof. Consider two incident switch trajectories sharing an end. Orient their period translators towards that end, so their heights are negative when the common end is negative, and take positive powers having equal heights. Far enough towards the shared end, any prescribed finite subwindows of the two trajectories lie on their common ray. Move the first subwindow there using its period translator and move back using the other translator. Both translators stabilize the original component because they preserve an edge in it. Their composite therefore belongs to A A A S S S S S S
For a fixed end, D z ∩ L D_z\cap L D z ∩ L A A A
[ h ( A ) : h ( D z ) ] [h(A):h(D_z)] [ h ( A ) : h ( D z )] S S S A A A [ h ( A ) : h ( Stab A ( τ ) ) ] [h(A):h(\operatorname{Stab}_A(\tau))] [ h ( A ) : h ( Stab A ( τ ))] S S S M τ M_{\tau} M τ
There are finitely many A A A A A A A A A S S S
Lemma 9.6 (Simultaneous incident free factors). At every vertex of every split action, the nontrivial incident edge groups, indexed by incidence-orbit representatives, are simultaneous free factors of its vertex group, up to separately chosen inner conjugations.
Proof. Fix a marked end ( z , e ) (z,e) ( z , e ) M = M z , e M=M_{z,e} M = M z , e e e e z z z M τ M_{\tau} M τ
In the core graph of the finite-path group K K K all switch lines through that path. They eventually embed jointly and disjointly modulo their K K K K K K M τ M_{\tau} M τ M τ M_{\tau} M τ
Disjoint embedded connected core subgraphs in a free graph give simultaneous free factors of its fundamental group, after choosing connecting paths. Therefore the indicated groups are simultaneous free factors of K K K M ≤ K M\le K M ≤ K K K K M M M
M \ K / M τ . M\backslash K/M_{\tau}. M \ K / M τ . Their pointwise groups are already contained in M M M M M M
Lemma 9.7 (A homology obstruction to a proper free quotient). Let F r F_r F r B ≤ L B\le L B ≤ L D L D_L D L
dim D L H 1 ( B ; D L ) ≤ r − 2. \dim_{D_L} H_1(B;D_L)\le r-2. dim D L H 1 ( B ; D L ) ≤ r − 2. In particular, it cannot surject onto a group containing a subgroup whose ambient homology image has dimension r − 1 r-1 r − 1
Proof. Choose a shortest nontrivial freely reduced word in the kernel. Its proper prefix vertices in the quotient are distinct; a repeated one would give a shorter nontrivial kernel word. Its closed labelled path has no cyclic backtrack, since cyclic cancellation would again shorten the kernel word. At least one evaluated Fox derivative is nonzero. More explicitly, cancellation of a contribution to a generator derivative would require an opposite traversal of the same oriented labelled edge. This repeats prefix vertices unless it is the forbidden backtrack. Thus this relator supplies a nonzero 2-boundary over the injective group ring inside D L D_L D L r r r B B B r − 1 r-1 r − 1 2 2 2
Proposition 9.8 (The component trees). Every component of the actual switch graph is a tree. For a nonisolated component with stabilizer A A A S = A ∩ L S=A\cap L S = A ∩ L r ≥ 2 r\geq2 r ≥ 2 S S S r r r
dim D L H 1 ( S ; D L ) = r − 1. \dim_{D_L} H_1(S;D_L)=r-1. dim D L H 1 ( S ; D L ) = r − 1. These approximants and S S S A A A
Proof. There are two steps to excluding circuits. Simultaneous incident free factors make the abstract incidence graph of groups free of a fixed rank r r r r − 1 r-1 r − 1
For separated marks, a split component is the tree C ( W ) \mathcal{C}(W) C ( W ) K ( W ) K(W) K ( W ) r = rk K ( W ) ≥ 2 r=\operatorname{rk}K(W)\geq2 r = rk K ( W ) ≥ 2 r r r
For a group acting without inversions on a connected graph, the incidence graph-of-groups fundamental group surjects onto the acting group, with kernel the ordinary fundamental group of the acted-on graph. This is seen by lifting the action to its universal covering tree. Apply this to a coarsened component. Its stabilizer contains an old K ( W ) K(W) K ( W ) H 1 ( L ; D L ) H_1(L;D_L) H 1 ( L ; D L ) r − 1 r-1 r − 1 r r r
Every finite circuit in the unsplit component would lift to a sufficiently coarsened split graph: move its finitely many shared-end marks far enough to make the necessary marks coincide. Since no split graph has a circuit, the unsplit component is a tree. Track split components containing one chosen switch line. Their stabilizers increase. Every s ∈ S s\in S s ∈ S s s s S S S r r r K ( W ) K(W) K ( W ) r − 1 r-1 r − 1
The quotient of the component tree by A A A D z D_z D z
1 ⟶ M τ ⟶ Stab A ( τ ) → h h ( Stab A ( τ ) ) ⟶ 1. 1\longrightarrow M_\tau\longrightarrow\operatorname{Stab}_A(\tau)\xrightarrow{h}h(\operatorname{Stab}_A(\tau))\longrightarrow1. 1 ⟶ M τ ⟶ Stab A ( τ ) h h ( Stab A ( τ )) ⟶ 1. Its height image is nonzero by periodicity, and M τ M_\tau M τ A A A
The period rule, Proposition 5.11, now applies to A A A 0 < β ( S ) = r − 1 < ∞ 0<\beta(S)=r-1<\infty 0 < β ( S ) = r − 1 < ∞ A A A G G G G G G A A A A A A
Finally, every end vertex group is a finite extension of an ascending free-group mapping torus. Every switch edge group has a finite-rank bilateral kernel and a nonzero height translator; the kernel is trivial or noncyclic. These vertex and edge groups are field-acyclic. One can compute this directly by the height homology sequence: over the skew Laurent division field, translation minus identity on the finite-dimensional kernel homology is invertible, as is seen from its leading Laurent degree. The same computation covers a cyclic edge group with trivial kernel. Finite-index passage preserves the vanishing. Mayer–Vietoris on the switch-tree splitting now gives field acyclicity of A A A
The initial captured system
Proposition 9.9 (Capture by periodic quasiconvex groups). Unless G G G G G G A A A S = A ∩ L S=A\cap L S = A ∩ L
(i) h ( A ) ≠ 0 h(A)\neq0 h ( A ) = 0 A A A
(ii) Each S S S
(iii) For representatives of the actual object orbits under L L L
⨁ H 1 ( S ; D L ) ⟶ H 1 ( L ; D L ) \bigoplus H_1(S;D_L)\longrightarrow H_1(L;D_L) ⨁ H 1 ( S ; D L ) ⟶ H 1 ( L ; D L ) are jointly injective. Distinct actual object stabilizers intersect in at most cyclic groups.
(iv) There is a uniform length such that the pointwise stabilizer of every longer noncyclically stabilized actual path, in T 0 T_0 T 0 T T T S S S
Proof. Use the components of C \mathcal{C} C D z D_z D z D z ∩ L D_z\cap L D z ∩ L
To obtain a simultaneous injection, choose finitely many free approximants from distinct L L L K ( W ) K(W) K ( W ) K ( W ) K(W) K ( W ) D L D_L D L
The resulting windows belong to distinct L L L H 1 ( L ; D L ) H_1(L;D_L) H 1 ( L ; D L )
We spell out the same-orbit intersection issue. Different L L L L L L S = ⋃ n F n S=\bigcup_n F_n S = ⋃ n F n r r r E = D S ⊗ Q S Q L E=D_S\otimes_{Q_S}Q_L E = D S ⊗ Q S Q L E E E D L D_L D L E E E F n F_n F n D L D_L D L F n F_n F n E E E L L L H 1 ( F n ; E ) → H 1 ( L ; E ) H_1(F_n;E)\to H_1(L;E) H 1 ( F n ; E ) → H 1 ( L ; E )
⨁ S g F n ∈ S \ L / F n H 1 ( S ∩ g F n g − 1 ; D S ) ↪ H 1 ( S ; D S ) . \bigoplus_{SgF_n\in S\backslash L/F_n} H_1(S\cap gF_ng^{-1};D_S)\hookrightarrow H_1(S;D_S). S g F n ∈ S \ L / F n ⨁ H 1 ( S ∩ g F n g − 1 ; D S ) ↪ H 1 ( S ; D S ) . This is the induced-coefficient restriction behind the slot rule; a dimension estimate alone would not suffice. It applies to the coarsened free approximants because their ambient homology maps have already been proved injective. The identity double coset already has dimension r − 1 = β ( S ) r-1=\beta(S) r − 1 = β ( S ) g F n g − 1 gF_ng^{-1} g F n g − 1 g ∉ S g\notin S g ∈ / S S ∩ g S g − 1 S\cap gSg^{-1} S ∩ g S g − 1 N L ( S ) = S N_L(S)=S N L ( S ) = S S S S
Now let C C C A A A C ∩ L C\cap L C ∩ L h ( C ) = 0 h(C)=0 h ( C ) = 0 h ( C ) ≠ 0 h(C)\ne0 h ( C ) = 0 C ∩ L C\cap L C ∩ L C C C C ∩ L = 1 C\cap L=1 C ∩ L = 1 C C C Z \mathbb{Z} Z
Finally, a sufficiently long noncyclically stabilized coarse path has a deep centered window W W W C ( W ) \mathcal{C}(W) C ( W ) T 0 T_0 T 0
If there are no active colors, Lemma 9.4 instead gives a uniform bound on lengths of noncyclically stabilized paths. The cyclic dictionary, Lemma 5.8, then bounds lengths of all nontrivially stabilized paths, so the refined action is acylindrical. The direct projection argument in Proposition 14.11, applicable with the already established hyperbolic rows and quasi-isometric edge maps, makes this a quasiconvex hierarchy. Its only potentially nonfree vertex is a hyperbolic root row, which itself splits over a quasiconvex cyclic subgroup with free and cyclic vertex groups. The hierarchy theorem therefore makes G G G
Cyclic closure of the periodic system
The captured periodic groups may still intersect cyclically. We first join these overlaps to obtain a malnormal system. The next two sections fold its actual subtrees into lines and enlarge groups when distinct lines share an end. Throughout, joint injection into H 1 ( L ; D L ) H_1(L;D_L) H 1 ( L ; D L ) k − 1 k-1 k − 1
We retain the ambient torsion-free hyperbolic primitive extension group G G G h : G → Z h:G\to\mathbb{Z} h : G → Z L L L T T T β ( V ) = dim D V H 1 ( V ; D V ) \beta(V)=\dim_{D_V}H_1(V;D_V) β ( V ) = dim D V H 1 ( V ; D V ) detected when one of its normalized Fox differences Δ ( a ) − Δ ( b ) \Delta(a)-\Delta(b) Δ ( a ) − Δ ( b ) H 1 ( L ; D L ) H_1(L;D_L) H 1 ( L ; D L )
The actual-row inputs are Lemmas 5.5, 5.10, 5.8, and 5.9, together with Propositions 5.6, 5.7, and 5.11. We also use the support and coset descent assertions of Theorem 6.1, the semilinear localization assertion of Lemma 7.1, and the captured periodic system of Proposition 9.9. In particular, the field homology of L L L k − 1 k-1 k − 1
Definition 10.1 (Admissible periodic system). An admissible system is a G G G G G G A A A
h ( A ) ≠ 0 h(A)\ne0 h ( A ) = 0 A A A G G G D A D_A D A
Stabilizers of distinct objects intersect at most cyclically.
For S = A ∩ L S=A\cap L S = A ∩ L 0 < β ( S ) < ∞ 0<\beta(S)<\infty 0 < β ( S ) < ∞ S S S
⨁ d ∈ L \ D H 1 ( S d ; D L ) ⟶ H 1 ( L ; D L ) \bigoplus_{d\in L\backslash D} H_1(S_d;D_L)\longrightarrow H_1(L;D_L) d ∈ L \ D ⨁ H 1 ( S d ; D L ) ⟶ H 1 ( L ; D L ) is injective. Representatives and coefficient transports are taken in the convention of Section 5.
There is a fixed length such that every noncyclic stabilizer of an actual path of that length lies wholly in one of the S d S_d S d
The system supplied by Proposition 9.9 is admissible. Its budget is
B ( D ) = ∑ d ∈ L \ D β ( S d ) ≤ k − 1. \mathcal{B}(\mathcal{D})=\sum_{d\in L\backslash D}\beta(S_d)\le k-1. B ( D ) = d ∈ L \ D ∑ β ( S d ) ≤ k − 1. This is a finite sum: each G G G L L L
We use the same closure operation after each end enlargement. Form a bipartite graph with the old objects as one class of vertices. The other class consists of maximal cyclic groups C < G C<G C < G C C C
Lemma 10.2 (The cyclic incidence forest). The cyclic incidence graph has finitely many G G G
Proof. Quasiconvex double-coset finiteness gives finitely many incidence-edge orbits. Suppose that a simple circuit has successive old vertices A i A_i A i C i C_i C i 1 ≠ c i ∈ C i ∩ A i ∩ A i + 1 1 \ne c_i \in C_i \cap A_i \cap A_{i+1} 1 = c i ∈ C i ∩ A i ∩ A i + 1 C i C_i C i c i c_i c i
Use literal Fox chains based at the identity, with the same word for c i c_i c i
E i = D A i ⊗ Q A i Q G = ∑ g ∈ G D A i g . E_i = D_{A_i} \otimes_{\mathbb{Q}A_i} \mathbb{Q}G = \sum_{g \in G} D_{A_i}g. E i = D A i ⊗ Q A i Q G = g ∈ G ∑ D A i g . The difference Δ ( c i ) − Δ ( c i − 1 ) \Delta(c_i) - \Delta(c_{i-1}) Δ ( c i ) − Δ ( c i − 1 ) E i E_i E i A i A_i A i b i b_i b i
∂ 2 b i = Δ ( c i ) − Δ ( c i − 1 ) , ∑ i b i = 0. \partial_2 b_i = \Delta(c_i) - \Delta(c_{i-1}), \qquad\sum_i b_i = 0. ∂ 2 b i = Δ ( c i ) − Δ ( c i − 1 ) , i ∑ b i = 0. The second equality follows because the boundaries telescope and the ambient two-boundary is injective.
Apply the single scalar-function map F \mathcal{F} F b i b_i b i D A i g D_{A_i}g D A i g A i A_i A i ∑ i F ( b i ) = 0 \sum_i \mathcal{F}(b_i) = 0 ∑ i F ( b i ) = 0 A j A_j A j A i ∩ A j A_i \cap A_j A i ∩ A j A i ∩ A j A_i \cap A_j A i ∩ A j
Old admissibility makes every A i ∩ A j A_i \cap A_j A i ∩ A j
To see the flux on one of these directions explicitly, write p n = F ( ( c i − 1 ) − 1 ) ( c i n ) p_n = \mathcal{F}((c_i - 1)^{-1})(c_i^n) p n = F (( c i − 1 ) − 1 ) ( c i n )
p n − 1 − p n = δ n , 0 . p_{n-1} - p_n = \delta_{n,0}. p n − 1 − p n = δ n , 0 . Thus the values on the two half-orbits are constants p − , p + p_-, p_+ p − , p + p − − p + = 1 p_- - p_+ = 1 p − − p + = 1 F ( Δ ( c i ) ) \mathcal{F}(\Delta(c_i)) F ( Δ ( c i )) c i c_i c i C i − 1 ≠ C i C_{i-1} \ne C_i C i − 1 = C i Δ ( c i − 1 ) \Delta(c_{i-1}) Δ ( c i − 1 ) F ( ( c i − 1 ) − 1 ) \mathcal{F}((c_i - 1)^{-1}) F (( c i − 1 ) − 1 )
Let R R R 2 R 2R 2 R F ( b i ) \mathcal{F}(b_i) F ( b i ) b i + b_i^+ b i +
∂ 2 b i + = the chosen nonzero constant half-ray of word paths + a finitely supported one-chain . \partial_2 b_i^+ = \text{the chosen nonzero constant half-ray of word paths} + \text{a finitely supported one-chain}. ∂ 2 b i + = the chosen nonzero constant half-ray of word paths + a finitely supported one-chain . The correction is finite because the Cayley complex is locally finite and the boundary matrices have finite propagation. The half-ray has divergence of nonzero total mass, whereas the divergence of any finitely supported one-chain has total mass zero. This contradicts ∂ 1 ∂ 2 b i + = 0 \partial_1\partial_2 b_i^+ = 0 ∂ 1 ∂ 2 b i + = 0
Lemma 10.3 (Component groups before the injection step). Let A ′ A' A ′ S ′ = A ′ ∩ L S'=A'\cap L S ′ = A ′ ∩ L A ′ A' A ′ G G G h ( A ′ ) ≠ 0 h(A')\ne0 h ( A ′ ) = 0 0 < β ( S ′ ) < ∞ 0<\beta(S')<\infty 0 < β ( S ′ ) < ∞ e + e_+ e + c + c_+ c + S ′ S' S ′
β ( S ′ ) = ∑ v ∈ S ′ \ O l d β ( S v ) + e + − c + . (38) \beta(S')=\sum_{v\in S'\backslash\mathrm{Old}}\beta(S_v)+e_+-c_+. \tag*{(38)} β ( S ′ ) = v ∈ S ′ \ Old ∑ β ( S v ) + e + − c + . (38) Here the old vertex groups are understood in their actual conjugate positions. No joint ambient injection for the new system is assumed in this lemma.
Proof. The finite quotient of the incidence forest makes A ′ A' A ′ h ( A ′ ) ≠ 0 h(A')\ne0 h ( A ′ ) = 0 A ′ A' A ′
For S ′ = A ′ ∩ L S'=A'\cap L S ′ = A ′ ∩ L H 0 H_0 H 0 H 1 H_1 H 1 S ′ S' S ′ H 0 H_0 H 0
β ( S ′ ) = ∑ v ∈ S ′ \ O l d β ( S v ) + e + − c + . \beta(S')=\sum_{v\in S'\backslash\mathrm{Old}}\beta(S_v)+e_+-c_+. β ( S ′ ) = v ∈ S ′ \ Old ∑ β ( S v ) + e + − c + . It is positive and finite. The period rule and the subtree gate criterion now give quasiconvexity of A ′ A' A ′ [66 ] .
We next control intersections; this must precede the new joint injection. For any two field-acyclic quasiconvex VCS groups B , C B,C B , C V V V
( ∑ g D B g ) ∩ ( ∑ g D C g ) = ∑ g D V g . \left(\sum_g D_Bg\right)\cap\left(\sum_g D_Cg\right)=\sum_g D_Vg. ( g ∑ D B g ) ∩ ( g ∑ D C g ) = g ∑ D V g . A cycle over the induced D V D_V D V H 1 ( V ; D V ) = 0 H_1(V;D_V)=0 H 1 ( V ; D V ) = 0 H 2 ( V ; D V ) = 0 H_2(V;D_V)=0 H 2 ( V ; D V ) = 0 H 0 = 0 H_0=0 H 0 = 0 V ≠ 1 V\ne1 V = 1
Apply this first to A ′ A' A ′
Finally the intersection of two distinct new component groups has at most cyclic old-node stabilizers. At an old node, any nontrivial such stabilizer lies in its unique maximal cyclic germ, and hence in the corresponding incidence stabilizer. Collapse the resulting nontrivial-edge stars. The intersection is free, and its field acyclicity again makes it at most cyclic. This supplies the intersection hypothesis for the next join forest without using the joint injection to be proved next. □ \square □
Lemma 10.4 (Detection under cyclic joins). Every noncyclic subgroup of a new height kernel S ′ S' S ′ H 1 ( S ′ ; D S ′ ) H_1(S';D_{S'}) H 1 ( S ′ ; D S ′ )
Proof. We first treat a horizontal piece and then the free-product joins. On the tree of S ′ S' S ′ H 0 H_0 H 0 H 1 H_1 H 1 H 1 H_1 H 1
Otherwise take two independent hyperbolics x , y x,y x , y v v v
Choose a nonzero c c c [ v , x v ] [v,xv] [ v , xv ] x c x − 1 xcx^{-1} x c x − 1 Δ ( c j ) \Delta(c_j) Δ ( c j ) [ v , x v ] [v,xv] [ v , xv ]
Δ ( x c x − 1 ) − Δ ( c ) = ( x − 1 ) ( Δ ( c ) − Δ ( x ) ) \Delta(xcx^{-1})-\Delta(c)=(x-1)(\Delta(c)-\Delta(x)) Δ ( x c x − 1 ) − Δ ( c ) = ( x − 1 ) ( Δ ( c ) − Δ ( x )) in affine chain coordinates modulo boundaries.
This writes Δ ( x ) \Delta(x) Δ ( x ) Δ ( c ) \Delta(c) Δ ( c ) y y y c ′ c' c ′ v v v Δ ( x ) − Δ ( y ) \Delta(x)-\Delta(y) Δ ( x ) − Δ ( y ) Δ ( c ) − Δ ( c ′ ) \Delta(c)-\Delta(c') Δ ( c ) − Δ ( c ′ ) ⟨ x ⟩ s S v \langle x\rangle sS_v ⟨ x ⟩ s S v ⟨ y ⟩ s S v \langle y\rangle sS_v ⟨ y ⟩ s S v S v S_v S v D S v D_{S_v} D S v Δ ( c ) = Δ ( c ′ ) \Delta(c)=\Delta(c') Δ ( c ) = Δ ( c ′ )
For the trivial-edge free-product splitting, an elliptic noncyclic subgroup uses the injected vertex homology. Otherwise its action is nonelementary: trivial edge stabilizers make an elementary nonelliptic subgroup cyclic. Choose two hyperbolics x , y x,y x , y q q q [ q , h q ] [q,hq] [ q , h q ]
Read the oriented half-edge E E E q q q x x x E , x E E,xE E , x E ⟨ x ⟩ E \langle x\rangle E ⟨ x ⟩ E [ q , x q ] [q,xq] [ q , x q ] ⟨ y ⟩ E \langle y\rangle E ⟨ y ⟩ E [ q , y q ] [q,yq] [ q , y q ] x x x − 1 -1 − 1 H 1 ( S ′ ) H_1(S') H 1 ( S ′ )
Proposition 10.5 (Joint injection after a cyclic join). For the new component system the map
⨁ i ∈ L ∖ D ′ H 1 ( S i ′ ; D L ) ⟶ H 1 ( L ; D L ) \bigoplus_{i\in L\setminus D'} H_1(S'_i;D_L)\longrightarrow H_1(L;D_L) i ∈ L ∖ D ′ ⨁ H 1 ( S i ′ ; D L ) ⟶ H 1 ( L ; D L ) is injective. Consequently a cyclic-join round preserves admissibility and does not decrease the budget. A round containing a positive-height cyclic join increases the budget by at least one.
Proof. Write D ′ D' D ′
V = ker ( ⨁ i ∈ L \ D ′ H 1 ( S i ′ ; D L ) ⟶ H 1 ( L ; D L ) ) . V=\ker\left(\bigoplus_{i\in L\backslash D'}H_1(S'_i;D_L)\longrightarrow H_1(L;D_L)\right). V = ker i ∈ L \ D ′ ⨁ H 1 ( S i ′ ; D L ) ⟶ H 1 ( L ; D L ) . There are three trees in the argument: the current join forest computes these homologies, the next join forest receives localized coefficients, and one level of the actual tree T T T
In the current Bass–Serre sequence, old joint injection makes the map from V V V H 0 H_0 H 0 L L L b b b i i i g i ∈ A i ′ g_i\in A'_i g i ∈ A i ′ h ( g i ) = b h(g_i)=b h ( g i ) = b t b g i − 1 t^b g_i^{-1} t b g i − 1 g g g b b b
p ⟼ t b p g − 1 . p\longmapsto t^bpg^{-1}. p ⟼ t b p g − 1 . Each one-dimensional coordinate is simple over the skew Laurent ring. Their finite direct sum is semisimple. If V ≠ 0 V\ne0 V = 0 1 1 1 t b g − 1 t^b g^{-1} t b g − 1
Choose an intrinsic D S i ′ D_{S'_i} D S i ′ B i B_i B i g i g_i g i λ i \lambda_i λ i
t b λ i g i − 1 B i = t b g i − 1 λ i , g λ i g i − 1 = λ i B i − 1 . t^b\lambda_i g_i^{-1}B_i=t^b g_i^{-1}\lambda_i,\qquad g\lambda_i g_i^{-1}=\lambda_iB_i^{-1}. t b λ i g i − 1 B i = t b g i − 1 λ i , g λ i g i − 1 = λ i B i − 1 . Lemma 7.1 localizes these rows to finitely many cosets whose transporters conjugate the contributing objects into one component of the next cyclic-join forest: their stabilizers share powers of g g g A ∗ A^* A ∗ S ∗ = A ∗ ∩ L S^*=A^*\cap L S ∗ = A ∗ ∩ L
For each nonzero row choose one of its height-zero transporters x i x_i x i x x x x x i − 1 ∈ S ∗ xx_i^{-1}\in S^* x x i − 1 ∈ S ∗ x i S i x i − 1 ≤ S ∗ x_iS_ix_i^{-1}\le S^* x i S i x i − 1 ≤ S ∗ x i x_i x i λ i x i − 1 \lambda_i x_i^{-1} λ i x i − 1 U ( S ∗ ) U(S^*) U ( S ∗ ) S ∗ S^* S ∗ L L L H 1 ( S ∗ ; U ( S ∗ ) ) H_1(S^*;U(S^*)) H 1 ( S ∗ ; U ( S ∗ ))
Use the period rule to represent each contributing intrinsic basis at one fixed actual P P P D S i ′ D_{S'_i} D S i ′ U ( S ∗ ) U(S^*) U ( S ∗ ) x i x_i x i a a a
Their ambient class is zero by the definition of V V V S ∗ S^* S ∗ U ( S ∗ ) ⋅ L U(S^*)\cdot L U ( S ∗ ) ⋅ L H 1 ( S ∗ ; U ( S ∗ ) ) H_1(S^*;U(S^*)) H 1 ( S ∗ ; U ( S ∗ )) V = 0 V=0 V = 0
At a positive-height cyclic node of degree m ≥ 2 m \ge2 m ≥ 2 m − 1 m-1 m − 1
Proposition 10.6 (Finite strict closure). Iterating cyclic joins stops after finitely many rounds. The final system is admissible and strictly malnormal: stabilizers of distinct objects have trivial intersection. Each is the full stabilizer of its object. For each final object d d d x x x X x ∩ S d ≠ 1 X_x \cap S_d \ne1 X x ∩ S d = 1 Z d Z_d Z d D d D_d D d D d D_d D d S d = D d ∩ L S_d = D_d \cap L S d = D d ∩ L
Proof. Every round containing a positive-height join raises the injected integer dimension, which is at most k − 1 k-1 k − 1
Follow a nested chain of component groups and put
A ∞ = ⋃ n A n , S ∞ = A ∞ ∩ L , M x ∞ = S ∞ ∩ X x . A^\infty= \bigcup_n A_n,\qquad S^\infty= A^\infty\cap L,\qquad M_x^\infty= S^\infty\cap X_x. A ∞ = n ⋃ A n , S ∞ = A ∞ ∩ L , M x ∞ = S ∞ ∩ X x . Homology commutes with this increasing union, and joint ambient injection gives 0 < β ( S ∞ ) ≤ k − 1 0 < \beta(S^\infty) \le k-1 0 < β ( S ∞ ) ≤ k − 1 h ( A ∞ ) ≠ 0 h(A^\infty) \ne0 h ( A ∞ ) = 0 Y Y Y M x ∞ M_x^\infty M x ∞ Y Y Y S ∞ S^\infty S ∞
Consider a finite run in Y Y Y M x ∞ M_x^\infty M x ∞ M x ∞ M_x^\infty M x ∞ c ∈ S ∞ c \in S^\infty c ∈ S ∞ ambient actual stabilizer K K K K K K n n n c c c n n n c c c K ≤ S ∞ K \le S^\infty K ≤ S ∞ M x ∞ M_x^\infty M x ∞
We now build a finite quotient of Y Y Y A ∞ A^\infty A ∞ M x ∞ M_x^\infty M x ∞ M x ∞ M_x^\infty M x ∞ X x X_x X x X x X_x X x A ∞ \ Y A^\infty\backslash Y A ∞ \ Y A ∞ A^\infty A ∞
A finite generating set of A ∞ A^\infty A ∞
For completeness, in this strict state let Y d Y_d Y d S d S_d S d x x x X x ∩ S d ≠ 1 X_x \cap S_d \ne1 X x ∩ S d = 1 x x x Y d Y_d Y d Y d Y_d Y d S d ′ S_{d'} S d ′ d ′ = d d'=d d ′ = d S d S_d S d Z d Z_d Z d
Strict row folding
Cyclic closure has produced a strict system and its actual subtrees Z d Z_d Z d Z d Z_d Z d
Subtrees and horizontal fibers
Lemma 11.1 (Root closure and the enlarged actual subtrees). In a strict system, each D d D_d D d G G G Z d Z_d Z d M x d = X x ∩ S d ≠ 1 M_x^d = X_x \cap S_d \ne1 M x d = X x ∩ S d = 1 D d D_d D d M x d M_x^d M x d S d S_d S d H 1 ( S d ) H_1(S_d) H 1 ( S d ) H 1 ( M x d ) H_1(M_x^d) H 1 ( M x d )
Proof. If g m ≠ 1 g^m \ne1 g m = 1 D d D_d D d g D d g − 1 = D g d gD_dg^{-1} = D_{gd} g D d g − 1 = D g d g d = d gd = d g d = d g ∈ D d g \in D_d g ∈ D d D d D_d D d Z d Z_d Z d M x d M_x^d M x d H 1 ( S d ) H_1(S_d) H 1 ( S d )
Equate points of each Z d Z_d Z d T T T T ^ \widehat{T} T L L L Z d Z_d Z d ℓ d \ell_d ℓ d
Indeed its image has one point per height, and any circuit in the quotient lifts after adding connecting paths in the Z d Z_d Z d
At a vertex or mid-edge level the components before identification are described by the bipartite graphs Γ w \Gamma_w Γ w x x x d d d w w w x ∈ Z d x \in Z_d x ∈ Z d w w w T ^ \widehat{T} T Y w = Stab G ( w ) ≤ L Y_w = \operatorname{Stab}_G(w) \le L Y w = Stab G ( w ) ≤ L Y w Y_w Y w L L L
Figure 3 shows the strict folding and its horizontal incidence data. The tree and homology assertions needed for this construction are proved below.
Figure 3. Strict folding is the transitive quotient generated by identifying same-height points within each Z d Z_d Z d T T T x , d x,d x , d x ∈ Z d x \in Z_d x ∈ Z d Γ w \Gamma_w Γ w
Lemma 11.2 (Raw horizontal jump vectors). Fix an actual row group X X X p X ⊂ L pX \subset L pX ⊂ L p X pX pX q X qX q X d d d q − p q-p q − p S d r S_dr S d r X X X d d d
Proof. The support assertion uses [6 ] . Fix an actual group X X X L L L p X pX pX L L L p X → q X pX \to qX pX → q X d d d 1 ≠ a ∈ S d ∩ p X p − 1 1 \ne a \in S_d \cap pXp^{-1} 1 = a ∈ S d ∩ pX p − 1 1 ≠ b ∈ S d ∩ q X q − 1 1 \ne b \in S_d \cap qXq^{-1} 1 = b ∈ S d ∩ q X q − 1 d ( q ) − d ( p ) d(q)-d(p) d ( q ) − d ( p )
q Δ X ( q − 1 b q ) − p Δ X ( p − 1 a p ) − ( Δ ( b ) − Δ ( a ) ) . q\Delta_X(q^{-1}bq) - p\Delta_X(p^{-1}ap) - \bigl(\Delta(b) - \Delta(a)\bigr). q Δ X ( q − 1 b q ) − p Δ X ( p − 1 a p ) − ( Δ ( b ) − Δ ( a ) ) . For the last parenthesis use an expression by cycles from the M x d M_x^d M x d D S d D_{S_d} D S d 2 2 2 S d r S_{dr} S d r X X X d d d q − p q-p q − p
The root-cylinder correction
Lemma 11.3 (Isolation and splitting of root cylinders). Suppose the actual row is
X = B ∗ ⟨ z ⟩ = ⟨ u a ⟩ ⟨ u ⟩ , a > 1 , X=B*_{\langle z\rangle=\langle u^a\rangle}\langle u\rangle,\qquad a>1, X = B ∗ ⟨ z ⟩ = ⟨ u a ⟩ ⟨ u ⟩ , a > 1 , where B B B z z z B B B U = ⟨ u ⟩ U=\langle u\rangle U = ⟨ u ⟩ α = 1 + u + ⋯ + u a − 1 \alpha=1+u+\cdots+u^{a-1} α = 1 + u + ⋯ + u a − 1
R = α e u − d B ( z ) . R=\alpha e_u-d_B(z). R = α e u − d B ( z ) . Let V = ∑ i V i V=\sum_i V_i V = ∑ i V i
V = P R in the division-field chain module (necessarily P = V u α − 1 ) , V=PR\quad\text{in the division-field chain module}\quad\text{(necessarily }P=Vu\alpha^{-1}\text{)}, V = P R in the division-field chain module (necessarily P = V u α − 1 ) , then, after applying the simultaneous scalar-function map, P P P R R R V i V_i V i X X X
Proof. The subgroup U U U X X X B B B z z z B B B u m u^m u m U U U
The internal root relation generates the kernel of the row chain quotient. Thus
V = P R , P = V u α − 1 , R = α e u − d B ( z ) . V=PR,\qquad P=Vu\alpha^{-1},\qquad R=\alpha e_u-d_B(z). V = P R , P = V u α − 1 , R = α e u − d B ( z ) . After applying the simultaneous scalar-function map, P P P U U U P P P X X X
Choose a ball radius large enough to contain every translate used in the fixed words for u a u^{a} u a z z z R R R U U U X X X U U U X X X U U U X X X
Write p n = P ( l u n ) p_n=P(lu^n) p n = P ( l u n ) V u = 0 V_u=0 V u = 0 u u u p n = p n − a p_n=p_{n-a} p n = p n − a B B B z z z B B B d B ( z ) d_B(z) d B ( z ) ⟨ z ⟩ \langle z\rangle ⟨ z ⟩ P ( l ) P(l) P ( l ) P ( l ) = 0 P(l)=0 P ( l ) = 0
After this support bound, write P = ∑ i P i + P 0 P=\sum_i P_i+P_0 P = ∑ i P i + P 0 P 0 P_0 P 0 P i P_i P i i i i R R R P R = ∑ i V i PR=\sum_i V_i P R = ∑ i V i V i − P i R V_i-P_iR V i − P i R
∂ ( P i R ) = P i ( α ( u − 1 ) − ( z − 1 ) ) = 0. \partial(P_iR)=P_i\bigl(\alpha(u-1)-(z-1)\bigr)=0. ∂ ( P i R ) = P i ( α ( u − 1 ) − ( z − 1 ) ) = 0. Consequently the correction preserves divergence. These are operations on scalar functions after projection; no multiplicativity of the scalar-function map for α − 1 \alpha^{-1} α − 1 X X X X X X
The folded incidence trees
Lemma 11.4 (Tree structure and homology of a folded row). For every folded place w w w Γ w \Gamma_w Γ w Y w Y_w Y w X x X_x X x S d S_d S d M x d M_x^d M x d
H 1 ( Y w ; D L ) → ≅ H 1 ( L ; D L ) . H_1(Y_w;D_L)\xrightarrow{\cong}H_1(L;D_L). H 1 ( Y w ; D L ) ≅ H 1 ( L ; D L ) . Proof. Sum these raw vectors around a putative simple circuit, fixing one frame per visited x x x D d D_d D d X X X X X X 1 1 1 − 1 -1 − 1
For a root row the raw sum around the circuit lies in the kernel of the row chain quotient and therefore is P R PR P R P = V u α − 1 P=V_u\alpha^{-1} P = V u α − 1 1 1 1 − 1 -1 − 1
The stabilizer and cocompactness assertions in the statement now follow by equivariance. The Bass–Serre Mayer–Vietoris sequence on Γ w \Gamma_w Γ w H 0 = 0 H_0=0 H 0 = 0 H 1 H_1 H 1 S d S_d S d X X X
Lemma 11.5 (Detection and separation at a folded level). Every noncyclic subgroup of Y w Y_w Y w
Proof. We verify detection: every noncyclic H ≤ Y w H \leq Y_w H ≤ Y w Δ ( h ) − Δ ( h ′ ) \Delta(h)-\Delta(h') Δ ( h ) − Δ ( h ′ ) Γ w \Gamma_w Γ w S d S_d S d X X X
In a root group use its own cyclic Bass tree. Elliptic noncyclic subgroups are in a free base with independent columns. Otherwise choose two hyperbolics with axes in different components off a pivot B B B
Write normalized paths from the base to its translates by using cyclic-amalgam normal forms and substituting d X ( u ) = α − 1 d B ( z ) d_X(u)=\alpha^{-1}d_B(z) d X ( u ) = α − 1 d B ( z ) h h h B B B U U U B B B U U U h i B h^iB h i B
The two incident cyclic stabilizers intersect trivially. Equality would force a function chain on that pivot coset with finite support and divergence of total mass one, impossible.
We next treat a subgroup that is nonelliptic on the folded incidence tree. If H H H Γ w \Gamma_w Γ w X X X d ( h ) d(h) d ( h ) d d d h X hX h X h h h ( h − 1 ) − 1 (h-1)^{-1} ( h − 1 ) − 1 X X X ⟨ h ⟩ \langle h\rangle ⟨ h ⟩ d d d D d D_d D d d d d
These are enclosing support patterns for sums/products, hence remain usable before any scalar function projection. In particular in the root case a further factor α − 1 \alpha^{-1} α − 1 U U U D d D_d D d P R PR P R P P P P P P R R R
This is impossible, in the free case directly so without correction.
Finally compare two distinct folded places. For different components at this level, any common subgroup cycle has expressions in the level basis supported inside their respective actual frame cosets by these same path formulas (in an arbitrary starting frame p p p h p hp h p d ( p ) d(p) d ( p ) X X X u u u □ \square □
Proposition 11.6 (Folded slot structure and restriction). *Fix a folded vertex v v v f f f K f K_f K f Γ f → Γ v \Gamma_f \to\Gamma_v Γ f → Γ v K f K_f K f Y f Y_f Y f d d d Γ v \Gamma_v Γ v K f K_f K f H ≤ Y v H \leq Y_v H ≤ Y v
⨁ f m o d H H 1 ( H ∩ Y f ; D L ) ⟶ H 1 ( H ; D L ) (39) \bigoplus_{f\ \mathrm{mod}\ H} H_1(H \cap Y_f; D_L) \longrightarrow H_1(H; D_L) \tag*{(39)} f mod H ⨁ H 1 ( H ∩ Y f ; D L ) ⟶ H 1 ( H ; D L ) (39) is injective.
Proof. Map each Γ f → Γ v \Gamma_f \to\Gamma_v Γ f → Γ v d d d K f K_f K f Y f Y_f Y f d d d Γ v \Gamma_v Γ v x , d x,d x , d M x d M_x^d M x d f f f x ∈ K f x \in K_f x ∈ K f X x X_x X x F ~ = X x ∩ Y f \widetilde{F}=X_x \cap Y_f F = X x ∩ Y f x x x f f f d d d Γ f \Gamma_f Γ f x x x d d d x x x Γ v \Gamma_v Γ v X x X_x X x f f f E E E E E E M x d M_x^d M x d M e d M_e^d M e d e e e F ~ \widetilde{F} F H 1 ( E ) → H 1 ( F ~ ) H_1(E) \to H_1(\widetilde{F}) H 1 ( E ) → H 1 ( F )
If X x X_x X x F ~ \widetilde{F} F k k k X x X_x X x H 1 ( E ) → H 1 ( X x ) H_1(E) \to H_1(X_x) H 1 ( E ) → H 1 ( X x ) X x X_x X x
For a root row, write its root splitting in the chosen slot as
X x = E ∗ ⟨ z ⟩ = ⟨ u a ⟩ ⟨ u ⟩ . X_x = E *_{\langle z\rangle=\langle u^a\rangle} \langle u\rangle. X x = E ∗ ⟨ z ⟩ = ⟨ u a ⟩ ⟨ u ⟩ . The actual edges of this slot correspond to the E E E S d S_d S d S d S_d S d S d S_d S d E E E d d d X x X_x X x x x x F ~ = X x \widetilde{F}=X_x F = X x
Suppose instead that no root power belongs to any S d S_d S d M = M x d M=M_x^d M = M x d E E E H 1 ( M ) H_1(M) H 1 ( M ) E E E M M M E E E E E E M M M
If M M M E E E H 1 ( M ) = 0 H_1(M)=0 H 1 ( M ) = 0 F ~ = E \widetilde{F}=E F = E
Both root alternatives give quasiconvexity and the needed comparison after restriction to an arbitrary subgroup of X x X_x X x F ~ = X x \widetilde{F}=X_x F = X x x x x F ~ = E \widetilde{F}=E F = E E E E
Compare now the Bass–Serre Mayer–Vietoris sequence for H H H Γ v \Gamma_v Γ v H ∩ Y f H \cap Y_f H ∩ Y f K f K_f K f d d d H 1 H_1 H 1 H 0 H_0 H 0 x x x H 1 H_1 H 1 (39) : a putative kernel first has zero projected edge-H 0 H_0 H 0 H 1 H_1 H 1
Corollary 11.7 (Strict folded tree). The quotient T ^ \widehat{T} T L L L Z d Z_d Z d ℓ d \ell_d ℓ d L L L
Proof. The quotient-tree and orbit assertions were proved in the construction. Start the homology assertion at one folded edge, using Lemma 11.4, and extend a path one edge at a time at either endpoint. Apply (39) with H H H
Enlargement at shared line ends
Distinct object stabilizers now intersect trivially, but their folded lines may still share an end. We replace such groups by full end stabilizers, prove that these remain quasiconvex and virtually compact special, and show that a genuine enlargement strictly increases the total injected homology dimension, which remains at most k − 1 k - 1 k − 1
End kernels and tail homology
Lemma 12.1 (End stabilizers). Suppose two distinct folded lines share an end. Choose the positive orientation if possible, reversing all orientations otherwise. The set of distinct positive ends ξ \xi ξ ℓ d \ell_d ℓ d G G G A ξ = Stab G ( ξ ) A_\xi= \operatorname{Stab}_G(\xi) A ξ = Stab G ( ξ ) S ξ = A ξ ∩ L S_\xi= A_\xi\cap L S ξ = A ξ ∩ L h ( A ξ ) ≠ 0 h(A_\xi) \ne0 h ( A ξ ) = 0 S ξ S_\xi S ξ ξ \xi ξ A ξ A_\xi A ξ
Proof. If some folded lines of distinct d d d ξ \xi ξ ℓ d \ell_d ℓ d A ξ A_\xi A ξ S ξ S_\xi S ξ S ξ S_\xi S ξ A ξ A_\xi A ξ
Lemma 12.2 (Joint injection for tails and ends). *For representatives of the L L L
⨁ ξ H 1 ( S ξ ; D L ) ⟶ H 1 ( L ; D L ) \bigoplus_{\xi} H_1(S_\xi;D_L) \longrightarrow H_1(L;D_L) ξ ⨁ H 1 ( S ξ ; D L ) ⟶ H 1 ( L ; D L ) is* is injective. The homology of every pointwise tail group K K K 0 < β ( S ξ ) < ∞ 0 < \beta(S_{\xi}) < \infty 0 < β ( S ξ ) < ∞
Proof. Choose rays to representatives of the L L L w w w K K K K n K_n K n n n n K n ↓ K K_n \downarrow K K n ↓ K
Work on the fixed incidence tree Γ w \Gamma_w Γ w d d d S d S_d S d ℓ d \ell_d ℓ d d d d S d S_d S d Z d Z_d Z d f f f K f K_f K f K f K_f K f K f K_f K f
In the Bass–Serre sequence for K K K Γ w \Gamma_w Γ w H 1 H_1 H 1 H 0 H_0 H 0 H 0 H_0 H 0 n n n n n n H 0 H_0 H 0 n n n
For the remaining actual-row part, use the free Cayley tree of each X x X_x X x K ∩ X x K \cap X_x K ∩ X x K n K_n K n n n n D L D_L D L
The three sequences now give injection on the original finite test space into the finite-prefix homologies. Those inject jointly into H 1 ( L ; D L ) H_1(L;D_L) H 1 ( L ; D L ) S ξ S_{\xi} S ξ k − 1 k-1 k − 1 S ξ S_{\xi} S ξ
Simultaneous boundary factors on a tail
Proposition 12.3 (The tail free-product decomposition). Let K K K w w w Γ w \Gamma_w Γ w K K K d 1 , … , d m d_1,\ldots,d_m d 1 , … , d m
K ≅ ( ∗ j = 1 m S d j ) ∗ F r , r < ∞ . (40) K \cong\left(\mathop{\ast}_{j=1}^{m} S_{d_j}\right) \ast F_r,\qquad r<\infty. \tag*{(40)} K ≅ ( ∗ j = 1 m S d j ) ∗ F r , r < ∞. (40) The displayed embeddings of full factors are their actual subgroup embeddings, up to the usual independent choices of conjugating basepoint paths.
Proof. Call an object node of Γ w \Gamma_w Γ w K K K S d S_d S d K K K K x = K ∩ X x K_x=K\cap X_x K x = K ∩ X x X x X_x X x
Remove the full nodes, and let B B B β ( B ) \beta(B) β ( B ) β ( K ) \beta(K) β ( K ) B B B β ( B ) < ∞ \beta(B)<\infty β ( B ) < ∞ B B B
We claim that the boundary incidence groups at this component are simultaneous free factors of B B B B B B B B B U ⊂ T U\subset T U ⊂ T B B B B ≤ L B\le L B ≤ L w ′ w' w ′
For an actual row y y y w ′ w' w ′ B y = B ∩ X y B_y=B\cap X_y B y = B ∩ X y y y y U U U y y y U U U B y B_y B y S d ′ S_{d'} S d ′ B y ≤ S d ′ B_y\le S_{d'} B y ≤ S d ′ y ∈ Z d ′ y\in Z_{d'} y ∈ Z d ′ B y B_y B y S d ′ S_{d'} S d ′ B y B_y B y B B B Γ w \Gamma_w Γ w
Identify the surviving object groups in the original Γ w \Gamma_w Γ w B ∩ S d ′ B\cap S_{d'} B ∩ S d ′ ℓ d ′ \ell_{d'} ℓ d ′ d ′ d' d ′ d ′ d' d ′ B ∩ S d ′ B\cap S_{d'} B ∩ S d ′ d ′ d' d ′ B B B d ′ d' d ′ M d ′ M_{d'} M d ′ X x X_x X x
Every original boundary incidence is retained in this argument, including cyclic ones. Its actual object label d ′ d' d ′ w w w w ′ w' w ′ B B B B B B B B B
Return to the splitting of K K K B B B
This eliminates each boundary factor against its image in the corresponding S d S_d S d
K ≅ ( ∗ j = 1 m S d j ) ∗ F r . K \cong\left(\mathop{*}_{j=1}^{m} S_{d_j}\right) * F_r. K ≅ ( ∗ j = 1 m S d j ) ∗ F r . The trivial-edge H 0 H_0 H 0
β ( K ) = ∑ j = 1 m β ( S d j ) + ( m − 1 ) + r . (41) \beta(K)=\sum_{j=1}^{m}\beta(S_{d_j})+(m-1)+r. \tag*{(41)} β ( K ) = j = 1 ∑ m β ( S d j ) + ( m − 1 ) + r . (41) Here m ≥ 1 m \ge1 m ≥ 1 β ( K ) \beta(K) β ( K ) m , r m,r m , r r r r
Virtually free ascending quotients
Filling the full factors in the tail decomposition (40) produces a finitely generated virtually free base. The induced ascending map may fail to be injective. We first remove this obstruction so that the ascending-torus theorem can be applied to the filled end groups.
Lemma 12.4 (Stabilizing a virtually free endomorphism). Let V V V ϕ : V → V \phi:V\to V ϕ : V → V ker ϕ n \ker\phi^n ker ϕ n
Proof. Choose a finite quotient q : V → Q q:V\to Q q : V → Q
F = ⋂ ψ : V → Q ker ψ . F=\bigcap_{\psi:V\to Q}\ker\psi. F = ψ : V → Q ⋂ ker ψ . There are finitely many such maps because V V V F F F ker q \ker q ker q ϕ n ( F ) \phi^n(F) ϕ n ( F ) n n n ϕ : ϕ n ( F ) → ϕ n + 1 ( F ) \phi:\phi^n(F)\to\phi^{n+1}(F) ϕ : ϕ n ( F ) → ϕ n + 1 ( F ) F F F V V V F F F [ V : F ] [V:F] [ V : F ]
If N = ker ϕ n N=\ker\phi^n N = ker ϕ n V / N ≅ ϕ n ( V ) V/N\cong\phi^n(V) V / N ≅ ϕ n ( V ) N N N V V V V / N V/N V / N
Now suppose ϕ \phi ϕ F 0 ≤ V F_0\le V F 0 ≤ V
F ∞ = ⋃ j ≥ 0 ϕ − j ( F 0 ) . F_\infty=\bigcup_{j\ge0}\phi^{-j}(F_0). F ∞ = j ≥ 0 ⋃ ϕ − j ( F 0 ) . The terms increase, are normal, and have index at most [ V : F 0 ] [V:F_0] [ V : F 0 ] ϕ − 1 ( F ∞ ) = F ∞ \phi^{-1}(F_\infty)=F_\infty ϕ − 1 ( F ∞ ) = F ∞ ϕ ( F ∞ ) ≤ F ∞ \phi(F_\infty)\le F_\infty ϕ ( F ∞ ) ≤ F ∞ V / F ∞ V/F_\infty V / F ∞ F ∞ F_\infty F ∞ F ∞ F_\infty F ∞
Proposition 12.5 (Quasiconvex and virtually compact special end groups). Each end stabilizer A ξ A_{\xi} A ξ is finitely generated, quasiconvex in G G G field-acyclic, and virtually compact special. It is torsion-free, locally indicable, coherent, of type F F F and of cohomological dimension at most two.
Proof. Choose σ \sigma σ p p p A ξ ′ = A ξ ∩ h − 1 ( p Z ) A'_{\xi}=A_{\xi}\cap h^{-1}(p\mathbb{Z}) A ξ ′ = A ξ ∩ h − 1 ( p Z ) K K K S ξ = ⋃ n ≥ 0 σ n K σ − n S_{\xi}=\bigcup_{n\geq0}\sigma^{n}K\sigma^{-n} S ξ = ⋃ n ≥ 0 σ n K σ − n A ξ ′ A'_{\xi} A ξ ′ K K K σ − 1 \sigma^{-1} σ − 1 K K K S d j S_{d_j} S d j S d j S_{d_j} S d j D d j ∩ A ξ ′ D_{d_j}\cap A'_{\xi} D d j ∩ A ξ ′ D d j D_{d_j} D d j σ \sigma σ A ξ ′ A'_{\xi} A ξ ′ A ξ A_{\xi} A ξ
Its height kernel has positive finite beta by tail injection, so the period rule and its actual-tree gate criterion make A ξ A_{\xi} A ξ G G G H 2 = 0 H_2=0 H 2 = 0 G G G [46 ] .
It remains to prove VCS. In A ξ ′ A'_{\xi} A ξ ′ D d ∩ A ξ ′ D_d\cap A'_{\xi} D d ∩ A ξ ′ ξ \xi ξ σ \sigma σ A ξ ′ A'_{\xi} A ξ ′
For each representative peripheral, choose a finite-index normal subgroup avoiding any prescribed finite set, and kill its intersection with S d S_d S d [55 ] .
The actual tail free-product decomposition identifies its filled base as
V = ( ∗ j = 1 m S d j / N d j ) ∗ F r , V=\left(*_{j=1}^{m}S_{d_j}/N_{d_j}\right)*F_r, V = ( ∗ j = 1 m S d j / N d j ) ∗ F r , a finitely generated virtually free group. Conjugation by σ − 1 \sigma^{-1} σ − 1 V V V A ξ ′ A'_{\xi} A ξ ′
Lemma 12.4 replaces this presentation by an injective virtually free ascending torus and then gives a finite-index free ascending torus. It is hyperbolic, so Theorem 4.1 makes the filled group VCS. We have obtained arbitrarily deep such fillings, with hyperbolic VCS peripheral images. Together with the finiteness and field hypotheses proved above, these are precisely the inputs to Theorem 3.1. Thus A ξ ′ A'_{\xi} A ξ ′ A ξ A_{\xi} A ξ
Termination of both closure operations
Proposition 12.6 (Strict growth at a shared end). Replacing the objects by distinct end stabilizers preserves all admissibility conditions. If some two distinct old lines share an end in the chosen orientation, this replacement strictly increases the budget. Subsequent cyclic closure does not decrease it.
Proof. Tail injection gives the new joint homology injection. To check detection, choose a finitely generated noncyclic subgroup of any noncyclic subgroup of S ξ S_{\xi} S ξ K K K K K K L L L
Fix a new end ξ \xi ξ L L L L L L S ξ S_{\xi} S ξ m m m
β ( S ξ ) ≥ ∑ j = 1 m β ( S d j ) + ( m − 1 ) . \beta(S_{\xi}) \ge\sum_{j=1}^{m}\beta(S_{d_j})+(m-1). β ( S ξ ) ≥ j = 1 ∑ m β ( S d j ) + ( m − 1 ) . This is strictly greater than the old sum when m ≥ 2 m \ge2 m ≥ 2
Suppose m = 1 m=1 m = 1 S d S_d S d S d S_d S d S ξ = S d S_{\xi}=S_d S ξ = S d S d S_d S d S ξ S_{\xi} S ξ
Sum over the new L L L T T T k − 1 k-1 k − 1
Theorem 12.7 (Terminal periodic system). Starting with the system of Proposition 9.9, finitely many cyclic-closure and end-enlargement rounds produce an admissible strict malnormal quasiconvex virtually compact special system { D d } \{D_d\} { D d } S d = D d ∩ L S_d=D_d\cap L S d = D d ∩ L T ^ \widehat{T} T ℓ d \ell_d ℓ d k − 1 k-1 k − 1
Proof. Use Proposition 10.6 to reach a strict state. Whenever an end is shared, apply the preceding construction in the relevant orientation, then restore strictness and fold again. All foldings use the same actual tree T T T k − 1 k-1 k − 1
Relative geometry of terminal rows
The terminal folding gives row groups assembled from actual rows and height kernels. We first control their peripheral incidences and build their relative geometry. The resulting comparison and intersection lemmas will supply the finite-window hypotheses for the terminal path argument, before any filling is chosen.
Standing terminal data
We work with the terminal system supplied by Theorem 12.7. We retain the actual tree T T T T ^ \widehat{T} T h : G → Z h:G\to\mathbb{Z} h : G → Z L = ker h L=\ker h L = ker h d d d D d D_d D d S d = D d ∩ L S_d=D_d\cap L S d = D d ∩ L Z d Z_d Z d ℓ d \ell_d ℓ d
(i) The groups D d D_d D d ℓ d \ell_d ℓ d
(ii) At a folded place w w w Γ w \Gamma_w Γ w x x x d d d X x X_x X x S d S_d S d M x d M^d_x M x d K f ⊂ Γ v K_f\subset\Gamma_v K f ⊂ Γ v F ~ x , f = X x ∩ Y f \widetilde{F}_{x,f}=X_x\cap Y_f F x , f = X x ∩ Y f X x X_x X x
(iii) There is a fixed actual capture length: the stabilizer of a sufficiently long actual segment, if noncyclic, lies wholly in one of the S d S_d S d Z d Z_d Z d
(iv) For the relative free-product trees constructed below we use the expansion and phase-uniqueness conclusions of the expansion-gate argument, with their finite-window hypotheses. We will verify those hypotheses before invoking those conclusions.
Throughout this section a nonparabolic subgroup means a nontrivial subgroup not contained in any actual S d S_d S d Y w Y_w Y w
Lemma 13.1 (Capture of a fixed monotone ray). If 1 ≠ c ∈ L 1\ne c\in L 1 = c ∈ L T ^ \widehat{T} T ℓ d \ell_d ℓ d c ∈ S d c\in S_d c ∈ S d
Proof. Take a remote infinite subray W W W W W W Γ \Gamma Γ W W W ℓ d ∩ W \ell_d\cap W ℓ d ∩ W
This incidence graph is a tree. There is a map from the cylinder complex to the graph: actual continuations and object intervals collapse to their vertex labels, place incidences map to the corresponding edges, and incidence strips have the same two labels at successive levels. Each prescribed vertex piece is connected. Thus a graph circuit could be lifted, using connecting paths inside these pieces, to a loop in the cylinder complex. Its image would be that circuit, contradicting simple connectivity.
The element c c c c c c c c c W W W c c c
A fixed object vertex gives c ∈ S d c\in S_d c ∈ S d T T T ⟨ c ⟩ \langle c\rangle ⟨ c ⟩ W W W c c c S d S_d S d W W W Z d Z_d Z d ℓ d \ell_d ℓ d W W W d d d W W W c ∈ S d c\in S_d c ∈ S d
Lemma 13.2 (Bounded peripheral incidences). There is a uniform constant C C C Y w ∩ S d ≠ 1 Y_w \cap S_d \ne1 Y w ∩ S d = 1 d Γ ^ ( w , ℓ d ) ≤ C d_{\widehat{\Gamma}}(w,\ell_d) \le C d Γ ( w , ℓ d ) ≤ C w ∉ ℓ d w \notin\ell_d w ∈ / ℓ d
Proof. Suppose Y w ∩ S d ≠ 1 Y_w \cap S_d \ne1 Y w ∩ S d = 1 ℓ d \ell_d ℓ d w w w v v v f f f d d d Γ v \Gamma_v Γ v K f K_f K f x x x
S d ∩ Y f = M x d ∩ F ~ x , f . S_d \cap Y_f = M_x^d \cap\widetilde{F}_{x,f}. S d ∩ Y f = M x d ∩ F x , f . This intersection is cyclic. Local cocompactness and quasiconvex double-coset finiteness in X x X_x X x ( d , v , f ) (d,v,f) ( d , v , f )
At every later off-line vertex, a nonzero element of this cyclic group acts hyperbolically on the incidence tree. Its axis determines at most one preserved continuation in each slot. Moreover a cyclic generator fixing the vertex fixes a continuation whenever a nonzero power does, by the stabilization argument in Lemma 12.2. Thus the path cannot reverse slots after departure. Unbounded distance for one of the finitely many departure types would give an off-line monotone ray fixed by a common nonzero power. Lemma 13.1 would put its end on ℓ d \ell_d ℓ d
The unfilled relative models
We now use these bounded incidence lists to construct the unfilled relative rows. The acting groups on adjacent places of T ^ \widehat{T} T Y f ≤ Y v Y_f \le Y_v Y f ≤ Y v Y z ∩ S d Y_z \cap S_d Y z ∩ S d Y z Y_z Y z S d S_d S d Γ v \Gamma_v Γ v M x M_x M x X X X K f ⊂ Γ v K_f \subset\Gamma_v K f ⊂ Γ v S d S_d S d X X X F ~ x \widetilde{F}_x F x X x X_x X x
Definition 13.3 (Exact relative embedding). Let J J J P \mathcal{P} P exact relative embedding of B ≤ J B \le J B ≤ J
(i) the nontrivial groups B ∩ p P p − 1 B \cap pPp^{-1} B ∩ pP p − 1 P ∈ P P \in\mathcal{P} P ∈ P B B B
(ii) B B B
(iii) its coned Cayley graph maps quasi-isometrically into the coned Cayley graph of J J J
Thus the apex map is injective on actual cosets. “Exact” refers to these induced intersections; it does not require that an induced subgroup equal the ambient peripheral.
Lemma 13.4 (Relative row geometry). Under the local hypotheses just listed, each folded vertex row Y v Y_v Y v Y f ↪ Y v Y_f \hookrightarrow Y_v Y f ↪ Y v
Proof. Each Y v Y_v Y v S d S_d S d X X X S d S_d S d X X X Γ v \Gamma_v Γ v
The infinite vertex stabilizers of this glued graph are exactly the included object groups, in their actual coset positions; ordinary vertices have finite stabilizers. Thus [24 ] gives a finite relative presentation with linear relative Dehn function. This comparison requires neither absolute finite generation of Y v Y_v Y v S d S_d S d
The edge group Y f Y_f Y f K f K_f K f F ~ x \widetilde{F}_x F x M M M S d S_d S d d ∈ Γ v − K f d \in\Gamma_v-K_f d ∈ Γ v − K f K f K_f K f d d d K f K_f K f
The models at X X X F ~ x \widetilde{F}_x F x X X X M M M
Summing distances over the point gluings preserves the relative embedding (each internal passage between distinct gluing apices costs positively). These models are equivalent to relative Cayley metrics by relative cocompactness or normal paths along the Bass tree.
External cyclic peripheral intersections to be added in Y v Y_v Y v
Thus adding them still gives relative hyperbolicity by elementary-subgroup peripheral enlargement [54 ] . For finitely many additions, iterate this statement over the distinct maximal cyclic types: a remaining type cannot become conjugate into an earlier one without being commensurable with it. The relative embedding above persists with the induced intersections, as we now prove separately.
Use coned Cayley graphs (ordinary T T T p α p_\alpha p α B ∩ p α P i p α − 1 B \cap p_\alpha P_i p_\alpha^{-1} B ∩ p α P i p α − 1 b b b b p α bp_\alpha b p α b p α P i bp_\alpha P_i b p α P i T T T
Recall the proper angle property at an old apex: for two incident ordinary vertices joined avoiding that apex in bounded length, the displacement belongs to a finite list inside the corresponding peripheral.
Project a new cyclic coset coarsely to the quasiconvex embedded orbit of B B B B B B
Frame vertices b b b a a a a a a b b b
Proper angles give b − 1 a b^{-1}a b − 1 a B B B
Conversely every infinite cyclic intersection gives unbounded old projection. This gives exactly the finite induced new list (malnormal loxodromic cyclic subgroups also in B B B B B B B = Y f B = Y_f B = Y f
Lemma 13.5 (Hyperbolicity of the partial metric). The tree of relative row spaces before filling is hyperbolic. Its metric is quasi-isometric to the partial Cayley metric on G G G S d S_d S d
Proof. Spelling paths in the Bass presentation identifies the tree of relative row spaces, up to quasi-isometry, with the partial Cayley metric that permits arbitrary moves in the S d S_d S d G G G S d S_d S d
Use a finite linear relative presentation of G G G D d D_d D d D d D_d D d D d / S d D_d/S_d D d / S d
Choose the filling with minimal ordinary relative area. A hole in a peripheral cluster can be filled entirely in that peripheral, by peripheral injectivity; doing so eliminates any ordinary cells inside. We may therefore treat clusters as discs. Each cluster boundary has a linear filling at a fixed Rips scale in its partial metric: push vertices by bounded moves to a lifted height line and fill there, or use the bounded metric for an extra cyclic. Combining these fillings gives a linear Rips filling for the partial Cayley graph, hence hyperbolicity. No finite generation of S d S_d S d
Unfilled comparisons and intersections
In these coned Cayley graphs, a bounded path avoiding an apex gives a peripheral displacement from a finite list, by proper angles. Add the finitely many subgroup offsets and endpoint connectors as ordinary generators first. This convention lets the following comparison retain actual subgroup-frame coordinates.
Lemma 13.6 (Unfilled comparison paths). Fix two exact relative embeddings into a row, and fixed ordinary endpoint connectors. Expanded intrinsic geodesics have the following comparison. Large-angle apices occur on both paths in the same order. Their corresponding inbound and outbound germs differ by peripheral elements from fixed finite lists. Between successive marked apices there are paired ordinary subgroup-frame checkpoints, at bounded successive path steps, whose discrepancies lie in a fixed finite subset of the row. The same conclusion holds in the central part of long paths tending to the endpoints of a common loxodromic element, with constants independent of remote endpoint costs.
Proof. Expand the two intrinsic geodesics in the row, including their fixed ordinary offsets. They are uniform quasigeodesics with no repeated ambient apices. Hyperbolicity makes them fellow travel coarsely in order, after adding the fixed endpoint connectors. In particular, matched positions sufficiently far apart occur in the same order.
Choose a large angle threshold. A turn above this threshold on one path must also occur on the other. Otherwise two short bridges, placed a fixed distance before and after the turn, together with the other path, give a bounded bypass avoiding its apex. Near an endpoint use the prescribed endpoint connector instead. The same bypasses bound the angles between the respective inbound germs, and between the outbound germs, at a common marked apex. Proper angles give their finite peripheral discrepancy lists.
The marked apices occur in the same order. A reversed pair could only occur in a bounded portion, by ordered fellow travelling. If their orders were v , w v,w v , w w , v w,v w , v w w w v v v v v v
Between successive marks all path turns have bounded angles. Pair ordinary subdivision points by short geodesic bridges, with bounded successive steps on each path. For a jump of a bridge at an apex z z z z z z z z z
All these constructions apply in long central portions of fellow-travelling paths without using their remote endpoint connectors. In particular they apply towards both endpoints of a common loxodromic element, with constants independent of those remote endpoint costs. Proper angles supply the finite lists; local compactness of the coned graph is not required. □ \square □
Lemma 13.7 (Unfilled intersections). If B , C B,C B , C J J J g ∈ J g \in J g ∈ J E = B ∩ g C g − 1 E=B\cap gCg^{-1} E = B ∩ g C g − 1 J J J B B B g C g − 1 gCg^{-1} g C g − 1
Proof. Put E = B ∩ g C g − 1 E=B\cap gCg^{-1} E = B ∩ g C g − 1 x ∈ E x\in E x ∈ E 1 , x 1,x 1 , x B B B g , g x g,gx g , g x g C gC g C u u u ( b , d ) ∈ B × g C (b,d)\in B\times gC ( b , d ) ∈ B × g C b − 1 d = u b^{-1}d=u b − 1 d = u ( h b u , h d u ) (hb_u,hd_u) ( h b u , h d u ) h ∈ E h\in E h ∈ E ( 1 , g ) (1,g) ( 1 , g ) h h h
At a marked jump the shared actual apex has forms h a = h ′ a ′ ha=h'a' ha = h ′ a ′ a , a ′ a,a' a , a ′ b u b_u b u h − 1 h ′ h^{-1}h' h − 1 h ′ E E E E E E J J J x x x
We next identify the peripheral structure, rather than just a list of elliptic subgroups. All nontrivial peripheral intersections appear: powers in any such intersection force arbitrarily large-angle passages at its apex, by the proper angle test using a fixed path towards it and its translate, since comparison paths then have bounded relative length. The apex map is injective on actual cosets after taking orbit representatives. The domain coned graph is fine by mapping bounded avoiding paths to avoiding paths (add the finitely many ordinary generators/offsets first), and is hyperbolic by quasigeodesicity. Fineness follows from proper angles at the domain apices (bounded avoiding paths give only finitely many next directions) and finite degree at ordinary vertices.
This graph has finite edge stabilizers and finitely many edge orbits; its infinite vertex stabilizers are exactly the nontrivial induced peripheral intersections, since the groups are torsion-free. Proposition 3.47 of [24 ] therefore gives relative hyperbolicity in the finite-relative-presentation sense, without assuming absolute finite generation of E E E B B B g C g − 1 gCg^{-1} g C g − 1
Corollary 13.8 (Unfilled nonparabolic double cosets). For fixed exact relative embeddings of B B B C C C J J J B g C BgC B g C B ∩ g C g − 1 B \cap gCg^{-1} B ∩ g C g − 1 J J J
Proof. A nontrivial nonparabolic subgroup of a torsion-free relatively hyperbolic group contains a loxodromic element. Compare long paths in the two embedded groups tending to its two endpoints. The central comparison in Lemma 13.6 gives paired frame points b ∈ B b \in B b ∈ B g c ∈ g C gc \in gC g c ∈ g C b − 1 g c b^{-1}gc b − 1 g c B g C BgC B g C g g g
Terminal paths and finite-path filling transfer
The unfilled relative models now supply the hypotheses for the expansion argument. We first obtain a terminal path bound, then fix the finite original tests that transfer path stabilizers exactly to a filling.
Relative free factors and the terminal path bound
Proposition 14.1 (Simultaneous relative factors). For a sufficiently long monotone folded path W W W H H H
H = ( ∗ j = 1 m ( H ∩ S d j ) ) ∗ F r . H = \left(\ast_{j=1}^{m}(H \cap S_{d_j})\right) \ast F_r. H = ( ∗ j = 1 m ( H ∩ S d j ) ) ∗ F r . The factors represent exactly the nontrivial peripheral intersections, up to H H H d d d
Proof. Let H = Stab ( W ) H = \operatorname{Stab}(W) H = Stab ( W ) H ∩ S d H \cap S_d H ∩ S d ℓ d \ell_d ℓ d W W W w , w ′ w,w' w , w ′
At an actual node x x x Γ w \Gamma_w Γ w H x = H ∩ X x H_x = H \cap X_x H x = H ∩ X x Γ w ′ \Gamma_{w'} Γ w ′ H x H_x H x x x x T T T S d S_d S d S d S_d S d
Every needed boundary incidence at x x x d d d w w w w ′ w' w ′ H x H_x H x H H H Γ w \Gamma_w Γ w
H ≅ ∗ j ( H ∩ S d j ) ∗ F r H \cong\mathop{\ast}_j (H \cap S_{d_j}) \ast F_r H ≅ ∗ j ( H ∩ S d j ) ∗ F r with all nontrivial peripheral intersections represented. Folded inertia bounds β ( H ) \beta(H) β ( H ) r r r H ≠ 1 H \ne1 H = 1
Use its relative Bass–Serre tree for R i R_i R i H H H d d d S d S_d S d S d S_d S d
Finally a nonparabolic subgroup has a loxodromic element in this tree. A nontrivial elliptic element fixes a unique vertex because edge stabilizers are trivial. Two elliptic elements with disjoint fixed sets have a loxodromic product. Hence a subgroup without loxodromics fixes one vertex, and exactness identifies its nontrivial stabilizer with an actual peripheral intersection. This would make the subgroup parabolic.
Choose a fixed short-window radius large enough for Proposition 14.1. We can now verify the expansion package of Definition 8.1 for the resulting relative trees. Condition (E1) is Lemma 13.5 together with the adjacent embeddings of Lemma 13.4. For (E2), intersect the current path group with each next edge group. Lemma 13.7 preserves exact relative embeddings, and Corollary 13.8 gives finitely many nonparabolic extensions. Induction over any fixed shape gives its finite list and uniform embedding constants, also with a specified common subwindow. Proposition 14.1 gives (E3), including exact peripheral restrictions and loxodromics for nonparabolic window groups. Finally distinct continuations have distinct terminal places at the same height; Lemma 11.5 makes their common stabilizer at most cyclic, which is (E4).
Proposition 14.2 (Terminal path bound). There is a length n 0 n_0 n 0 T ^ \widehat{T} T n 0 n_0 n 0 S d S_d S d
Proof. We first rule out arbitrarily long monotone paths with nonparabolic stabilizers. Suppose they exist. Normalize a centered short window to one fixed W 0 W_0 W 0 H 0 = Stab ( W 0 ) H_0 = \operatorname{Stab}(W_0) H 0 = Stab ( W 0 ) R 0 R_0 R 0 R 0 R_0 R 0 H 0 H_0 H 0 W 0 W_0 W 0
At the same time, diagonalize the oriented turn labels of the axes, indexed relative to the marked edge. At each index the label is eventually constant or leaves every finite set of labels. We call the latter an escaping turn. A turn at a finite-valence trivial vertex cannot escape.
There cannot be two escaping turns. Choose consecutive escaping indices, so all intermediate turn labels are constant, and normalize the intervening edge path. Its edge markings determine actual coordinates because its edge stabilizers are trivial. Its two end vertices are then fixed peripheral vertices labeled by two distinct actual objects d , e d,e d , e S d S_d S d S e S_e S e
Consequently at least one half of the axis coordinates has no escaping turn. The constant turns and the marked edge define an actual infinite ray ρ ⊂ R 0 \rho\subset R_0 ρ ⊂ R 0 ρ \rho ρ τ \tau τ
The first diagonal argument has produced an infinite carried ray. It has not yet produced a gate event that recurs on that ray. We obtain repeated finite arcs by a second normalization, now of disjoint blocks. Choose infinitely many disjoint blocks in ρ \rho ρ 8 R + 4 8R+4 8 R + 4 2 R 2R 2 R H 0 H_0 H 0 2 R 2R 2 R H 0 H_0 H 0 2 R 2R 2 R ρ \rho ρ
Fix a depth large enough for phase uniqueness in both signs. The corresponding window stabilizer has finite edge quotient on its minimal subtree. Among the repeated arcs, infinitely many marked initial edges belong to one orbit under this smaller stabilizer. Its transporter carries the entire arc, since the arcs already have the same H 0 H_0 H 0 τ \tau τ τ \tau τ
Each transporter has height zero and fixes that initial window. Preserving its expanding end therefore makes it fix the ray to that end pointwise. Lemma 13.1 puts every nonidentity transporter in the same S d S_d S d H 0 ∩ S d H_0 \cap S_d H 0 ∩ S d R 0 R_0 R 0 ρ \rho ρ
It follows that sufficiently long paths in T ^ \widehat{T} T S d S_d S d
Corollary 14.3 (The additional root cyclic). In the root case in which no power of the root cyclic group is claimed by the object system, its maximal cyclic overgroup may be added to the peripheral family. Its nontrivial incidences have finite orbit lists, and the relative embeddings and finite-path conclusions continue to hold with the enlarged induced systems.
Proof. In the root case when powers of actual U = ⟨ u ⟩ U = \langle u\rangle U = ⟨ u ⟩ S d S_d S d U U U G G G U ∗ U^* U ∗ T ^ \widehat{T} T Γ \Gamma Γ x x x K f K_f K f f f f X x X_x X x F ~ \widetilde{F} F
The finite tests and their order of choice
For all filling assertions in this section, choose for each periodic peripheral a normal subgroup N d ⊴ D d N_d \trianglelefteq D_d N d ⊴ D d N d ≤ S d N_d \leq S_d N d ≤ S d [ S d : N d ] < ∞ [S_d:N_d] < \infty [ S d : N d ] < ∞ G G G P = Y w ∩ S d P = Y_w \cap S_d P = Y w ∩ S d N P = P ∩ N d N_P = P \cap N_d N P = P ∩ N d P / N P P/N_P P / N P D d / N d D_d/N_d D d / N d compatible below includes these restrictions and normality conditions. Their construction, and attainment of the further finite requests below, belong to Section 15.
We record the finite-window and quotient argument more precisely. Use relative embeddings as above into one row J J J B , C ≤ J B,C \leq J B , C ≤ J [55 ] . Thus in coned quotient graphs hyperbolicity constants are uniform and bounded avoiding paths at an apex give displacement represented by a fixed finite original list in its peripheral (isolated component lemma applied to the uniformly linear presentations with fixed relative relators). This holds also with any fixed finitely many ordinary letters added. Here and below fixed elements needed in ordinary connectors (even in groups only assumed relatively finitely generated) can be added as ordinary letters first. Fixed finite word equalities lift exactly, as does membership of a fixed finite element in a specified peripheral (add the tested element as an ordinary edge to use the angle bound, then finite-list injection). These are relative statements not requiring finite generation inside the peripherals. We induce standalone fillings on B , C B,C B , C
The finite peripheral membership safeguards and induced subgroup fillings below have an antecedent in Groves–Manning’s H H H [25 ] [Definitions 3.1–3.3 and Propositions 4.5–4.6]. Here the rows are only relatively finitely generated, with no finite-generation requirement inside the peripherals, and we impose additional product-membership and exact-intersection conditions. The stronger exact-intersection assertion below is therefore a separate local argument, not a theorem invoked from that source.
Definition 14.4 (Peripheral compatibility tests). Fix the rows and subgroup embeddings to be tested, and a finite list of their peripheral subgroup occurrences. For each ambient peripheral P P P N P ⊲ P N_P \lhd P N P ⊲ P
inside a peripheral, finite-list membership in each indicated intersection subgroup Q Q Q Q t R QtR QtR t t t Q Q Q R R R
Q ‾ ∩ s ˉ R ‾ s ˉ − 1 = Q ∩ s R s − 1 \overline{Q} \cap\bar{s}\overline{R}\bar{s}^{-1} = Q \cap sRs^{-1} Q ∩ s ˉ R s ˉ − 1 = Q ∩ s R s − 1 s s s
Here all letters and subgroups are in that one peripheral’s coordinates. The second condition concerns only cyclic pairs with trivial intersection (commensurate cycles satisfy it automatically, by the gcd formula for subgroups in cyclic quotients). Full here means the entire indicated height-zero peripheral (not the whole periodic group). In Bass applications the property full-or-cyclic for new path intersections follows since at each constituent place a nontrivial object intersection is itself full or cyclic. We explain simultaneous attainment separately, including preserving the required double-coset inequalities by separability around cyclics.
The word “specified” has a fixed quantifier order. First choose the original subgroup embeddings and a path length to be tested. Next choose the geometric comparison constants and the finite original connector alphabets. Next record the resulting finite membership, product-membership and exact-intersection requests. Only then choose the kernels. The construction in Section 15 attains every such finite collection. Increasing depth later preserves previously imposed finite exclusions.
For membership, the assertion is that an element of the specified original finite list that belongs to Q Q Q Q Q Q Q t R QtR QtR
Lemma 14.5 (Induced subgroup fillings). For a fixed finite collection of exact relative embeddings into rows, one can choose the finite tests in Definition 14.4 so that every sufficiently long filling satisfying them has the following properties. Each standalone induced subgroup filling injects into the row filling, and the resulting coned embeddings have common quasi-isometry constants independent of the filling. Their expanded intrinsic geodesics have no repeated ambient peripheral apex.
Proof. We prove the assertion for B B B ( λ 0 , ϵ 0 ) (\lambda_0,\epsilon_0) ( λ 0 , ϵ 0 ) ( λ 1 , ϵ 1 ) (\lambda_1,\epsilon_1) ( λ 1 , ϵ 1 ) λ 1 ϵ 1 \lambda_1\epsilon_1 λ 1 ϵ 1
A bounded-length intrinsic coned geodesic in B ˉ \bar{B} B ˉ B B B B ‾ \overline{B} B
Next compare an ordinary-endpoint such path, starting at identity, with any ambient geodesic shortcut. In the bounded union graph a jump not at a cut apex separating start and end has bounded angle (connect inbound to start and outbound to end, and these to each other avoiding that apex). Thus lift such letters as just above: the actual finite-list representative in an intersection subgroup after the membership test gives even the required standalone peripheral value, since its kernel there is the restriction and the reduced intersection subgroup injects in the ambient peripheral reduction.
Cut apices occur on both paths in the same order. Discrepancies between their ordinary inbound germs, likewise outbound germs, are from finite lists in that peripheral by avoiding paths through start or end. Lift those connectors. Between cuts all consistency equations with the connectors hold by fixed-list injection. Choose any actual lift in the indicated subgroup for the B B B
Thus some actual B B B ( λ 1 , ϵ 1 ) (\lambda_1,\epsilon_1) ( λ 1 , ϵ 1 )
A repeated apex on the global expanded path would cut off a subpath of length at most λ 1 ϵ 1 \lambda_1\epsilon_1 λ 1 ϵ 1
Lemma 14.6 (Finite original comparison data). For the fixed embeddings and connectors of Lemma 13.6, every sufficiently long filling satisfying Lemma 14.5 has the same comparison conclusions. All discrepancy and peripheral-germ lists are images of fixed finite lists in the original groups, independent of the filling.
Proof. The subgroup-filling lemma gives uniform quasi-isometry constants and no repeated ambient apices for expanded intrinsic geodesics. Relative filling gives a uniform hyperbolicity constant. At every fixed bypass length, the isolated-component bound places peripheral displacements in the image of a fixed finite original list. This remains true after the finitely many connector letters and offsets have been added as ordinary generators.
Apply the proof of Lemma 13.6 with these constants. Its fellow-travelling, marked-apex, and bypass arguments use only those bounds. Every bounded-angle jump in a comparison bridge consequently has a representative in its fixed original list. Products of the bounded number of such letters give fixed original checkpoint-discrepancy lists as well. The central comparison uses the same constants independently of remote endpoints, exactly as before filling.
Corollary 14.7 (Finite double-coset representatives). For fixed original exact relative embeddings of B , C B,C B , C Proof. Compare paths towards the two endpoints of the common loxodromic as in Corollary 13.8, now using Lemma 14.6. A pair of central frame points gives the required representative from its fixed original discrepancy list. Row peripheral images are finite, so every infinite-order element relevant here is loxodromic.
Proposition 14.8 (Exact intersection transfer). Fix exact relative embeddings B , C ↪ J B,C\hookrightarrow J B , C ↪ J g ∈ J g\in J g ∈ J
B ˉ ∩ g ˉ C ˉ g ˉ − 1 = B ∩ g C g − 1 ‾ . \bar{B}\cap\bar{g}\bar{C}\bar{g}^{-1}=\overline{B\cap gCg^{-1}}. B ˉ ∩ g ˉ C ˉ g ˉ − 1 = B ∩ g C g − 1 . The notation B ˉ , C ˉ \bar{B},\bar{C} B ˉ , C ˉ
Proof. Use the quotient comparison of Lemma 14.6. Its small-letter and germ-connector lists determine the finite tests before kernels are chosen. We show how their local lifts concatenate. For an element x ˉ \bar{x} x ˉ 1 1 1 x ˉ \bar{x} x ˉ B ˉ \bar{B} B ˉ g ˉ \bar{g} g ˉ x ˉ g ˉ \bar{x}\bar{g} x ˉ g ˉ g ˉ C ˉ \bar{g}\bar{C} g ˉ C ˉ
d ˉ i = b ˉ i u ˉ i , u ˉ 0 = u ˉ m = g ˉ . \bar{d}_i=\bar{b}_i\bar{u}_i,\qquad\bar{u}_0=\bar{u}_m=\bar{g}. d ˉ i = b ˉ i u ˉ i , u ˉ 0 = u ˉ m = g ˉ . Choose once a lift u i u_i u i g g g
β ˉ i u ˉ i + 1 = u ˉ i γ ˉ i , β ˉ i ∈ B ˉ , γ ˉ i ∈ C ˉ . \bar{\beta}_i\bar{u}_{i+1}=\bar{u}_i\bar{\gamma}_i,\qquad\bar{\beta}_i\in\bar{B},\quad\bar{\gamma}_i\in\bar{C}. β ˉ i u ˉ i + 1 = u ˉ i γ ˉ i , β ˉ i ∈ B ˉ , γ ˉ i ∈ C ˉ . On a small strip, the increments have lifts β i ∈ B \beta_i\in B β i ∈ B γ i ∈ C \gamma_i\in C γ i ∈ C
β i u i + 1 = u i γ i . \beta_i u_{i+1}=u_i\gamma_i. β i u i + 1 = u i γ i . At a marked strip, remove the fixed entry and exit offsets. The remaining equation is exactly q t = s r qt=sr q t = sr s ˉ ∈ Q ˉ t ˉ R ˉ \bar{s}\in\bar{Q}\bar{t}\bar{R} s ˉ ∈ Q ˉ t ˉ R ˉ s s s q 0 t = s r 0 q_0t=sr_0 q 0 t = s r 0 q ˉ , r ˉ \bar{q},\bar{r} q ˉ , r ˉ
q ˉ q ˉ 0 − 1 ∈ Q ˉ ∩ s ˉ R ˉ s ˉ − 1 . \bar{q}\bar{q}_0^{-1}\in\bar{Q}\cap\bar{s}\bar{R}\bar{s}^{-1}. q ˉ q ˉ 0 − 1 ∈ Q ˉ ∩ s ˉ R ˉ s ˉ − 1 . Choose a ∈ Q ∩ s R s − 1 a\in Q\cap sRs^{-1} a ∈ Q ∩ s R s − 1
q = a q 0 , r = s − 1 a s r 0 q=aq_0,\qquad r=s^{-1}asr_0 q = a q 0 , r = s − 1 a s r 0 solve the original equation with exactly the prescribed reduced values. Restoring the entry and exit offsets gives (14.1) on this strip too. Telescoping all strips now gives
( β 0 ⋯ β m − 1 ) g = g ( γ 0 ⋯ γ m − 1 ) . (\beta_0\cdots\beta_{m-1})g=g(\gamma_0\cdots\gamma_{m-1}). ( β 0 ⋯ β m − 1 ) g = g ( γ 0 ⋯ γ m − 1 ) . The left product belongs to B ∩ g C g − 1 B\cap gCg^{-1} B ∩ g C g − 1 x ˉ \bar{x} x ˉ
Transfer along Bass–Serre paths
Theorem 14.9 (Finite-path transfer). Fix an integer n n n
(i) The induced row fillings and edge fillings form the quotient graph of groups for G ˉ \bar{G} G ˉ
(ii) Every reduced path of length at most n n n
(iii) Taking n n n
Here the ambient filled group is hyperbolic by sufficiently long filling of the original relatively hyperbolic pair: the periodic peripheral reductions are finite-by-cyclic and hence hyperbolic, and the additional cyclic reductions are finite. The conclusion about ambient quasiconvexity is proved separately below.
Proof. Before filling, Lemma 13.7 gives exact relative embeddings for finite-path stabilizers into the incident rows. Corollary 13.8 gives finitely many nonparabolic extensions of each fixed path shape. We choose the filling tests recursively, for paths of length at most n n n
At each step the two groups to intersect are the current actual path stabilizer and the next edge group, both embedded in the current terminal row. Fix their original quasi-isometry constants first. Corollary 14.7 then gives a finite original list of possible extension representatives downstairs whenever the intersection is infinite. Only after fixing that list do we impose Proposition 14.8 for its representatives, and Lemma 14.5 for the resulting actual intersection groups. Repeating for the fixed length n n n
Under these tests the induced row and edge fillings give a quotient graph of groups. Its Bass presentation is exactly that of G ˉ \bar{G} G ˉ
Now let a reduced path downstairs have length at most n n n
Each filled row is ordinarily hyperbolic. Its finite relative generating set and its finitely many finite peripheral images give a finite generating set. Insert presentations of those finite groups into a linear relative presentation. In this fixed quotient each peripheral portion has bounded linear ordinary filling cost, yielding an ordinary linear isoperimetric inequality. This argument requires no absolute finite generation of the original height kernels. The induced relative embeddings give quasi-isometric edge maps.
Take n n n n n n G ‾ \overline{G} G n n n
Otherwise choose sufficiently many disjoint oriented length-n n n c c c c c c ⟨ c ⟩ \langle c\rangle ⟨ c ⟩
Finally Proposition 14.11 below promotes the row and edge embeddings to ambient quasiconvexity. This is the additional conclusion needed for the hierarchy after cubulating the filled vertex groups.
Lemma 14.10 (Projection of quasiconvex cosets). Fix finitely many quasiconvex subgroup types in a hyperbolic group J J J R , C R,C R , C H , K H,K H , K g K gK g K H H H g K gK g K C C C h ∈ H , k ∈ K h \in H,k \in K h ∈ H , k ∈ K
w = h − 1 g k , ∣ w ∣ ≤ R . w=h^{-1}gk,\qquad|w|\le R. w = h − 1 g k , ∣ w ∣ ≤ R . After translation by h − 1 h^{-1} h − 1 C C C ( H ∩ w K w − 1 ) w (H\cap wKw^{-1})w ( H ∩ w K w − 1 ) w
Proof. Thin quadrilaterals give the first alternative when the distance between the two quasiconvex cosets is sufficiently large. More explicitly, a long interval between two projection points must, away from bounded end segments, fellow travel the side joining the two original points. Quasiconvexity puts that side near H H H H , g K H,gK H , g K R R R ( h , g k ) (h,gk) ( h , g k ) h − 1 h^{-1} h − 1 H H H g K gK g K w K wK w K w w w
For the normalized close pair ( 1 , w ) (1,w) ( 1 , w ) w K ∩ N L ( H ) wK\cap N_L(H) w K ∩ N L ( H ) L L L I = H ∩ w K w − 1 I=H\cap wKw^{-1} I = H ∩ w K w − 1
( a , b ) ∈ H × w K , a − 1 b = d , ∣ d ∣ ≤ L (a,b)\in H\times wK,\qquad a^{-1}b=d,\qquad|d|\le L ( a , b ) ∈ H × w K , a − 1 b = d , ∣ d ∣ ≤ L with a fixed discrepancy d d d I I I a ′ a − 1 = b ′ b − 1 ∈ I a'a^{-1}=b'b^{-1}\in I a ′ a − 1 = b ′ b − 1 ∈ I d d d w w w I w Iw I w H , K , w H,K,w H , K , w I I I
Proposition 14.11 (Quasiconvexity from bounded segment stabilizers). Let A A A n n n n n n A A A
Proof. Use the ordinary Bass–Serre tree of Cayley spaces and fix a target vertex row. Project toward it, one row at a time: take coarse nearest projection to the quasiconvex parent-edge coset, then transfer across the edge by its quasi-isometric identification. Each such map is coarsely Lipschitz, and successive maps agree up to bounded error across the attaching strips. We first work at a fixed bounded depth.
After the first transfer the projected set lies in an attaching-edge orbit. The inductive assertion is this: after j j j j j j
For the next transfer apply Lemma 14.10 to the controlling coset and the next attaching coset. A bounded projection gives the first alternative and remains bounded under further transfers of bounded depth. Otherwise normalize a close pair. Only finitely many bounded discrepancies occur, and the new controlling group is exactly the intersection of the previous path stabilizer with the next edge stabilizer. In the notation of that lemma, frame at w w w I w Iw I w w − 1 I w ≤ K w^{-1}Iw \le K w − 1 I w ≤ K
At depth n + 1 n+1 n + 1 n n n normalized groups now gives a uniform bound on their orbit diameters. This is where the normalization is essential: a bound on finite subgroup orders alone would not bound the diameter of an arbitrarily based conjugate orbit. It follows that the projection of every entire row at depth n + 1 n+1 n + 1
Extend over each farther branch by a constant chosen from the image of its depth-n + 1 n+1 n + 1
Choice of the height-fiber kernels
We construct normal kernels that preserve height, make each height-kernel image finite, and meet the transfer tests already fixed. Their restrictions to actual incidence groups must also give the exact homological graph covers used in the later cubulation. The construction therefore combines finite-index separations with characters and their nontrivial cyclic-cover cohomology modes.
Fix one periodic peripheral D D D S = D ∩ L S=D\cap L S = D ∩ L Z D ⊂ T Z_D\subset T Z D ⊂ T D D D G G G h ( D ) ≠ 0 h(D)\ne0 h ( D ) = 0 Z D Z_D Z D
M ξ = D ξ = S ∩ X ξ M_{\xi}=D_{\xi}=S\cap X_{\xi} M ξ = D ξ = S ∩ X ξ is nontrivial, finitely generated, and of type F F F X ξ X_{\xi} X ξ
All extra requirements in this section form finite lists up to the original group action. They include prescribed finite avoidance, cyclic membership and double-coset tests from Section 14, every actual incidence stabilizer, any additional required nontrivial groups, specified cyclic generators in S S S D D D H ⊴ D H \trianglelefteq D H ⊴ D N N N N N N
Local cones and finite-index preparations
Lemma 15.1 (The finite family of local cones). For a vertex x ∈ Z D x \in Z_D x ∈ Z D
⨁ e in the slot at x m o d M x Q D ⊗ Q M e C ∗ ( M e ) ⟶ Q D ⊗ Q M x C ∗ ( M x ) \bigoplus_{\substack{e\ \text{in the slot at }x\\ \bmod M_x}} \mathbb{Q}D \otimes_{\mathbb{Q}M_e} C_*(M_e) \longrightarrow\mathbb{Q}D \otimes_{\mathbb{Q}M_x} C_*(M_x) e in the slot at x mod M x ⨁ Q D ⊗ Q M e C ∗ ( M e ) ⟶ Q D ⊗ Q M x C ∗ ( M x ) has a finite based free mapping cone over Q D \mathbb{Q}D Q D D D D_D D D D D D
Proof. Choose finite classifying complexes for the stabilizers, using their already established torsion-free hyperbolicity and [46 ] , and cellular maps realizing their inclusions, with the actual attaching conjugations. The orbit list at each vertex and the list of vertex types are finite. Induction of their cellular chains therefore gives the asserted finite free complexes with rational group-ring entries.
In degree one the induced map is an isomorphism by the local period rule, because every incident row in the slot participates. In degree zero all terms vanish over D D D_D D D g g g g − 1 g-1 g − 1 D D D_D D D
Lemma 15.2 (Novikov openness after restriction). There is a normal finite-index subgroup H 0 ⊴ D H_0 \trianglelefteq D H 0 ⊴ D
O 0 ⊂ Hom ( H 0 , R ) \mathcal{O}_0 \subset\operatorname{Hom}(H_0,\mathbb{R}) O 0 ⊂ Hom ( H 0 , R ) on which all the restricted local cones admit contractions over the ordinary rational group Novikov ring. This property persists after passing to any further finite-index subgroup, with a possibly new nonempty open cone.
Proof. A virtually compact special group is virtually RFRS [28 , 1 ] . Apply the finite-complex form of the RFRS Novikov theorem [36 ] to the direct sum of the cones in Lemma 15.1, after restriction to a finite-index RFRS subgroup. The subgroup-field compatibility gives the required field acyclicity after restriction. Take the normal core of the resulting subgroup.
Here are the restriction and openness details, which also justify later finite-index refinements. In finite coset coordinates, a Novikov contraction restricts to a matrix contraction over the completion for the restricted character. The finitely many coset coordinate shifts do not affect summability. Given such a contraction s s s s 0 s_0 s 0
E = 1 − d s 0 − s 0 d E = 1 - ds_0 - s_0d E = 1 − d s 0 − s 0 d has strictly positive character value. This is possible because the differentials have finite support. Positivity of these finitely many terms remains true in a neighborhood of the character. On that neighborhood, ( 1 − E ) − 1 = ∑ n > 0 E n (1-E)^{-1} = \sum_{n>0} E^n ( 1 − E ) − 1 = ∑ n > 0 E n E E E d d d s 0 ( 1 − E ) − 1 s_0(1-E)^{-1} s 0 ( 1 − E ) − 1 □ \square □
Lemma 15.3 (Visible and independent evaluations). After a further normal finite-index refinement of H 0 H_0 H 0 M M M u M ∈ M ∩ H 0 u_M \in M \cap H_0 u M ∈ M ∩ H 0 H 0 H_0 H 0 ⟨ a ⟩ , ⟨ b ⟩ \langle a\rangle,\langle b\rangle ⟨ a ⟩ , ⟨ b ⟩ a , b a,b a , b H 0 H_0 H 0
Hom ( H 0 , Q ) ⟶ Q 2 . \operatorname{Hom}(H_0,\mathbb{Q}) \longrightarrow\mathbb{Q}^2. Hom ( H 0 , Q ) ⟶ Q 2 . The conditions hold for all the conjugated tests as well.
Proof. Choose one nonidentity element in each required M M M D D D D D D
Pass first to a compact special subgroup of finite index. Quasiconvex subgroups of a hyperbolic compact special group are virtual retracts [28 , 29 ] . Apply this to suitable powers of each chosen cyclic group and each of the rank-two groups. A retraction onto a cyclic group gives its nonzero rational character; a retraction onto the rank-two group gives the two coordinate characters. Intersect the finitely many resulting subgroups and take a normal core in D D D □ \square □
We may simultaneously make the finite avoidance and girth requirements part of H 0 H_0 H 0 D D D [47 ] . For a failed membership test u ∉ Q t R u \notin Q t R u ∈ / QtR M M M M M M M ≤ D M \le D M ≤ D
Lemma 15.4 (Separating actual cyclic pairs). There is a normal finite-index subgroup H ⊴ D H \trianglelefteq D H ⊴ D H 0 H_0 H 0 ⟨ a ⟩ , ⟨ b ⟩ \langle a\rangle, \langle b\rangle ⟨ a ⟩ , ⟨ b ⟩ D / H D/H D / H
Proof. List the conjugacy types of the maximal cyclic groups containing the specified generators. In a torsion-free hyperbolic group these form a malnormal quasiconvex family. Apply the malnormal special quotient theorem [3 ] with sufficiently deep power kernels in these maximal cyclic groups, retaining its designated finite-index normal subgroups P ˙ i ⊴ P i \dot{P}_i \trianglelefteq P_i P ˙ i ⊴ P i [55 ] . We obtain a hyperbolic virtually compact special quotient D ′ D' D ′ P i = ⟨ c i ⟩ P_i = \langle c_i\rangle P i = ⟨ c i ⟩ [ P i : P ˙ i ] [P_i:\dot{P}_i] [ P i : P ˙ i ] [ P i : P i ∩ H 0 ] [P_i:P_i \cap H_0] [ P i : P i ∩ H 0 ] P ˙ i \dot{P}_i P ˙ i H 0 H_0 H 0 a = c k a = c^k a = c k c c c c c c ∣ k ∣ r |k|r ∣ k ∣ r a a a r r r
We verify separation in D ′ D' D ′ [55 ] ; hence they are chosen before the power exponents. Preserve the distinctness of the finitely many marked apex cosets by the finite peripheral membership tests. Let v ≠ w v \ne w v = w I I I u ∈ I u \in I u ∈ I γ \gamma γ v v v w w w u γ u\gamma u γ u u u w w w v v v
Normalize v v v P = ⟨ c ⟩ P = \langle c\rangle P = ⟨ c ⟩ x ∈ P ‾ x \in\overline{P} x ∈ P u γ u\gamma u γ x − 1 u x = u x^{-1}ux = u x − 1 ux = u P ‾ \overline{P} P u ∈ I u \in I u ∈ I c e c^e c e ∣ e ∣ ≤ B |e| \le B ∣ e ∣ ≤ B ∣ P ‾ ∣ = n > 3 B |\overline{P}| = n > 3B ∣ P ∣ = n > 3 B P ‾ \overline{P} P n n n n / 3 n/3 n /3 I = 1 I = 1 I = 1 D ′ D' D ′
Finally use residual finiteness of D ′ D' D ′ A , B ≤ D ′ A,B \le D' A , B ≤ D ′ x y − 1 xy^{-1} x y − 1 x ∈ A x \in A x ∈ A y ∈ B y \in B y ∈ B D / H 0 D/H_0 D / H 0 D ′ D' D ′ H 0 H_0 H 0 D D D H H H
Freeze H H H O ⊂ Hom ( H , R ) \mathcal{O} \subset\operatorname{Hom}(H,\mathbb{R}) O ⊂ Hom ( H , R ) D − D- D − D D D
S 0 = H ∩ S , M 0 = M ∩ H . S_0 = H \cap S,\qquad M^0 = M \cap H. S 0 = H ∩ S , M 0 = M ∩ H . Integral characters and scalar local systems
Lemma 15.5 (A finite generic character family). There is a nonempty finite family of integral characters χ j : H → Z \chi_j : H \to\mathbb{Z} χ j : H → Z
(i) every χ j \chi_j χ j M 0 M^0 M 0 H H H
(ii) for each specified pair a , b a,b a , b m a , m b m_a,m_b m a , m b H H H
( χ j ( a m a ) ) j , ( χ j ( b m b ) ) j \left(\chi_j\left(a^{m_a}\right)\right)_j,\qquad\left(\chi_j\left(b^{m_b}\right)\right)_j ( χ j ( a m a ) ) j , ( χ j ( b m b ) ) j are linearly independent over Q \mathbb{Q} Q
(iii) the family is closed under conjugation by D D D
There are arbitrarily large odd primes ℓ \ell ℓ ℓ \ell ℓ
Proof. The forbidden zero evaluations are finitely many proper rational hyperplanes by Lemma 15.3. For each pair, the evaluation map to R 2 \mathbb{R}^2 R 2 O \mathcal{O} O
Impose the same conditions on the finitely many conjugated tests, then add all D D D H H H ℓ \ell ℓ
Lemma 15.6 (Scalar specialization). One can choose such an odd prime ℓ \ell ℓ F \mathbb{F} F ℓ \ell ℓ ℓ \ell ℓ j j j λ ℓ = 1 \lambda^\ell=1 λ ℓ = 1 λ ≠ 1 \lambda\ne1 λ = 1
H 1 ( M x 0 ; F λ χ j ) ⟶ ⨁ e in the chosen slot at x m o d M x 0 H 1 ( M e 0 ; F λ χ j ) H^1(M_x^0;\mathbb{F}_{\lambda^{\chi_j}})\longrightarrow
\bigoplus_{\substack{e\text{ in the chosen slot at }x\\ \bmod M_x^0}}
H^1(M_e^0;\mathbb{F}_{\lambda^{\chi_j}}) H 1 ( M x 0 ; F λ χ j ) ⟶ e in the chosen slot at x mod M x 0 ⨁ H 1 ( M e 0 ; F λ χ j ) are isomorphisms. Actual attaching conjugations are included in this notation. All these local groups have zero H 0 H^0 H 0
Proof. For an integral character χ : H → Z \chi:H\to\mathbb{Z} χ : H → Z
Q H ^ χ ⟶ Q ( ( t ) ) , ∑ g ∈ H r g g ⟼ ∑ n ∈ Z ( ∑ χ ( g ) = n r g ) t n . \widehat{\mathbb{Q}H}^{\chi}\longrightarrow\mathbb{Q}((t)),\qquad
\sum_{g\in H}r_g g\longmapsto
\sum_{n\in\mathbb{Z}}\left(\sum_{\chi(g)=n}r_g\right)t^n. Q H χ ⟶ Q (( t )) , g ∈ H ∑ r g g ⟼ n ∈ Z ∑ χ ( g ) = n ∑ r g t n . Each inner sum is finite and the occurring heights are bounded below. The same support condition makes every coefficient of a product a finite sum, so this is indeed a ring homomorphism. Apply it to a Novikov contraction of each restricted cone. The resulting finite complex is contractible over Q ( ( t ) ) \mathbb{Q}((t)) Q (( t )) Q [ t , t − 1 ] \mathbb{Q}[t,t^{-1}] Q [ t , t − 1 ] Q ( t ) \mathbb{Q}(t) Q ( t ) Q ( ( t ) ) \mathbb{Q}((t)) Q (( t )) Q ( t ) \mathbb{Q}(t) Q ( t )
Choose a rational-function contraction and clear all its denominators. For the finitely many cones and characters this gives a nonzero Laurent polynomial f ( t ) ∈ Z [ t , t − 1 ] f(t) \in\mathbb{Z}[t,t^{-1}] f ( t ) ∈ Z [ t , t − 1 ] f ( t ) ≠ 0 f(t) \ne0 f ( t ) = 0 f ( t − 1 ) f(t^{-1}) f ( t − 1 ) ℓ \ell ℓ ℓ \ell ℓ Q \mathbb{Q} Q Φ ℓ ( t ) \Phi_\ell(t) Φ ℓ ( t ) ℓ \ell ℓ ℓ \ell ℓ
For completeness, the local maps asserted in the statement are exactly the maps controlled by these cones. Restricting an induced complex from D D D H H H H \ D / M x H\backslash D/M_x H \ D / M x H ∩ M x H \cap M_x H ∩ M x H H H T \mathbb{T} T χ j \chi_j χ j ℓ \ell ℓ λ χ j \lambda^{\chi_j} λ χ j ℓ \ell ℓ
Proposition 15.7 (Restriction to one actual level). For every j j j λ \lambda λ r r r Z D Z_D Z D
H 1 ( S 0 ; T λ χ j ) ⟶ ∏ ξ at level r mod S 0 H 1 ( M ξ 0 ; T λ χ j ) . (42) H^1(S_0;\mathbb{T}_{\lambda\chi_j}) \longrightarrow\prod_{\substack{\xi\ \text{at level }r\\ \operatorname{mod} S_0}} H^1(M^0_\xi;\mathbb{T}_{\lambda\chi_j}). \tag*{(42)} H 1 ( S 0 ; T λ χ j ) ⟶ ξ at level r mod S 0 ∏ H 1 ( M ξ 0 ; T λ χ j ) . (42) The product on the right is finite. In particular the group on the left is finite-dimensional, and restriction to any individual actual stabilizer is onto.
Proof. The H H H S 0 S_0 S 0 S 0 S_0 S 0 D D D S S S D D D S S S S 0 S_0 S 0 S S S
Take products of the local restriction isomorphisms over all vertex orbits at one level. On either slot the incident-edge indexing is exactly the S 0 S_0 S 0
The cohomological Mayer–Vietoris sequence for the action on the tree Z D Z_D Z D [14 , 59 ] . All its local degree-zero terms vanish. Its degree-one part therefore identifies H 1 ( S 0 ; T λ χ j ) H^1(S_0;\mathbb{T}_{\lambda\chi_j}) H 1 ( S 0 ; T λ χ j )
∏ ξ vertex mod S 0 H 1 ( M ξ 0 ; T λ χ j ) ⟶ ∏ ξ edge mod S 0 H 1 ( M ξ 0 ; T λ χ j ) , \prod_{\xi\ \text{vertex mod }S_0} H^1(M^0_\xi;\mathbb{T}_{\lambda\chi_j})
\longrightarrow
\prod_{\xi\ \text{edge mod }S_0} H^1(M^0_\xi;\mathbb{T}_{\lambda\chi_j}), ξ vertex mod S 0 ∏ H 1 ( M ξ 0 ; T λ χ j ) ⟶ ξ edge mod S 0 ∏ H 1 ( M ξ 0 ; T λ χ j ) , where the map is the difference of the two endpoint restrictions. Prescribing the data at one level determines the data at successive levels uniquely, in both directions, by the preceding isomorphisms. Every such prescription gives a compatible product. This proves (K1); it also explains why infinite products over the whole height line cause no extra degrees of freedom.
The exact kernel and its graph covers
For every j j j
S ℓ j = ker ( S 0 → χ j Z → Z / ℓ ) , M ℓ j = M 0 ∩ S ℓ j . S_{\ell j}=\ker\left(S_0\xrightarrow{\chi_j}\mathbb{Z}\to\mathbb{Z}/\ell\right),\qquad M_{\ell j}=M^0\cap S_{\ell j}. S ℓ j = ker ( S 0 χ j Z → Z / ℓ ) , M ℓ j = M 0 ∩ S ℓ j . The selected nonzero evaluations imply that both S 0 S_0 S 0 M 0 M^0 M 0 T \mathbb{T} T δ \delta δ
e 0 = 1 ℓ ∑ r = 0 ℓ − 1 δ r , e_0=\frac{1}{\ell}\sum_{r=0}^{\ell-1}\delta^r, e 0 = ℓ 1 r = 0 ∑ ℓ − 1 δ r , V j = ker ( e 0 : H 1 ( S ℓ j ; T ) → H 1 ( S ℓ j ; T ) ) , V_j=\ker\left(e_0:H^1(S_{\ell j};\mathbb{T})\to H^1(S_{\ell j};\mathbb{T})\right), V j = ker ( e 0 : H 1 ( S ℓ j ; T ) → H 1 ( S ℓ j ; T ) ) , V M j = ker ( e 0 : H 1 ( M ℓ j ; T ) → H 1 ( M ℓ j ; T ) ) . V_{M_j}=\ker\left(e_0:H^1(M_{\ell j};\mathbb{T})\to H^1(M_{\ell j};\mathbb{T})\right). V M j = ker ( e 0 : H 1 ( M ℓ j ; T ) → H 1 ( M ℓ j ; T ) ) . Thus these spaces contain all, and only, the nontrivial deck modes.
Proposition 15.8 (Exact local annihilators). Each V j V_j V j V M j V_{M_j} V M j M M M ker V j = ⋂ v ∈ V j ker v \ker V_j=\bigcap_{v\in V_j}\ker v ker V j = ⋂ v ∈ V j ker v
N = ⋂ j ( S ℓ j ∩ ker V j ) (43) N=\bigcap_j\left(S_{\ell j}\cap\ker V_j\right) \tag*{(43)} N = j ⋂ ( S ℓ j ∩ ker V j ) (43) is normal in D D D S S S
M ∩ N = ⋂ j ( M ℓ j ∩ ⋂ v ∈ V M j ker v ) . (44) M\cap N=\bigcap_j\left(M_{\ell j}\cap\bigcap_{v\in V_{M_j}}\ker v\right). \tag*{(44)} M ∩ N = j ⋂ M ℓ j ∩ v ∈ V M j ⋂ ker v . (44) The equality uses no simultaneous surjectivity across different j j j
Proof. Finite-index Shapiro and the decomposition of the regular T [ Z / ℓ ] \mathbb{T}[\mathbb{Z}/\ell] T [ Z / ℓ ]
H 1 ( S ℓ j ; T ) ≅ ⨁ λ ℓ = 1 H 1 ( S 0 ; T λ χ j ) . H^1(S_{\ell j};\mathbb{T})\cong\bigoplus_{\lambda^\ell=1}H^1(S_0;\mathbb{T}_{\lambda\chi_j}). H 1 ( S ℓ j ; T ) ≅ λ ℓ = 1 ⨁ H 1 ( S 0 ; T λ χ j ) . This remains valid when S 0 S_0 S 0 ℓ \ell ℓ V j V_j V j
Since M 0 M^0 M 0 M ℓ j M_{\ell j} M ℓ j
H 1 ( S 0 ; T λ χ j ) ⟶ H 1 ( M 0 ; T λ χ j ) , H^1(S_0;\mathbb{T}_{\lambda\chi_j})\longrightarrow H^1(M^0;\mathbb{T}_{\lambda\chi_j}), H 1 ( S 0 ; T λ χ j ) ⟶ H 1 ( M 0 ; T λ χ j ) , which is onto by Proposition 15.7. Restriction is deck-equivariant (a deck generator may be represented inside M 0 M^0 M 0 V j V_j V j V M j V_{M_j} V M j
With trivial coefficients, first cohomology is Hom ( S ℓ j , T + ) \operatorname{Hom}(S_{\ell j},\mathbb{T}^+) Hom ( S ℓ j , T + ) V j V_j V j ker V j \ker V_j ker V j S ℓ j S_{\ell j} S ℓ j j j j [ S : N ] < ∞ [S:N]<\infty [ S : N ] < ∞ D D D S ℓ j S_{\ell j} S ℓ j N N N V j → V M j V_j\to V_{M_j} V j → V M j j j j
Corollary 15.9 (The outgoing graph-cover property). Let M M M A component L H L_H L H M 0 M^0 M 0 ℓ \ell ℓ L ℓ j L_{\ell j} L ℓ j M ℓ j M_{\ell j} M ℓ j L N L_N L N M ∩ N M \cap N M ∩ N L ℓ j L_{\ell j} L ℓ j F \mathbb{F} F char F ≠ ℓ \operatorname{char}\mathbb{F} \ne\ell char F = ℓ
In particular, if e 1 , e 2 e_1,e_2 e 1 , e 2 L ℓ j L_{\ell j} L ℓ j L H L_H L H L N L_N L N e 1 ∗ − e 2 ∗ e_1^* - e_2^* e 1 ∗ − e 2 ∗
Proof. Connectedness of L ℓ j L_{\ell j} L ℓ j χ j : M 0 → Z / ℓ \chi_j:M^0 \to\mathbb{Z}/\ell χ j : M 0 → Z / ℓ (44) , using H 1 ( L ℓ j ; F ) = Hom ( M ℓ j , F + ) H^1(L_{\ell j};\mathbb{F}) = \operatorname{Hom}(M_{\ell j},\mathbb{F}^+) H 1 ( L ℓ j ; F ) = Hom ( M ℓ j , F + ) H H H e 1 ∗ − e 2 ∗ e_1^* - e_2^* e 1 ∗ − e 2 ∗ (44) therefore annihilates all its periods on L N L_N L N
Proposition 15.10 (Orders and exact cyclic intersections). For every specified generator a ∈ S a \in S a ∈ S m a m_a m a D / H D/H D / H Its order in S / N S/N S / N m a ℓ m_a\ell m a ℓ S / N S/N S / N
Proof. Put z = a m a ∈ S 0 z=a^{m_a}\in S_0 z = a m a ∈ S 0 χ j ( z ) \chi_j(z) χ j ( z ) ℓ \ell ℓ z z z S 0 / S ℓ j S_0/S_{\ell j} S 0 / S ℓ j z ℓ z^\ell z ℓ S ℓ j S_{\ell j} S ℓ j z z z v ∈ V j v\in V_j v ∈ V j v ( z ℓ ) = λ v ( z ℓ ) v(z^\ell)=\lambda v(z^\ell) v ( z ℓ ) = λ v ( z ℓ ) λ ≠ 1 \lambda\ne1 λ = 1 v ( z ℓ ) = 0 v(z^\ell)=0 v ( z ℓ ) = 0 V j V_j V j z ℓ ∈ ker V j z^\ell\in\ker V_j z ℓ ∈ ker V j j j j a m a ℓ ∈ N a^{m_a\ell}\in N a m a ℓ ∈ N a q ∈ N ⊂ H a^q\in N\subset H a q ∈ N ⊂ H q = m a r q=m_ar q = m a r a q ∈ S ℓ j a^q\in S_{\ell j} a q ∈ S ℓ j j j j r χ j ( z ) = 0 ( m o d ℓ ) r\chi_j(z)=0\pmod{\ell} r χ j ( z ) = 0 ( mod ℓ ) ℓ ∣ r \ell\mid r ℓ ∣ r
Suppose next that a q N = b r N a^qN=b^rN a q N = b r N D / H D/H D / H q = m a u q=m_au q = m a u r = m b v r=m_bv r = m b v a q b − r ∈ S ℓ j a^qb^{-r}\in S_{\ell j} a q b − r ∈ S ℓ j j j j
u ( χ j ( a m a ) ) j − v ( χ j ( b m b ) ) j = 0 over F ℓ . u\bigl(\chi_j(a^{m_a})\bigr)_j-v\bigl(\chi_j(b^{m_b})\bigr)_j=0\quad\text{over }\mathbb{F}_\ell. u ( χ j ( a m a ) ) j − v ( χ j ( b m b ) ) j = 0 over F ℓ . The two vectors are independent, so ℓ ∣ u , v \ell\mid u,v ℓ ∣ u , v a q , b r ∈ N a^q,b^r\in N a q , b r ∈ N
Remark 15.11 (Additional maximal cyclic peripherals). If an added peripheral is itself P = ⟨ c ⟩ P=\langle c\rangle P = ⟨ c ⟩ H P = ⟨ c r ⟩ H_P=\langle c^r\rangle H P = ⟨ c r ⟩ r r r χ ( c r ) = 1 \chi(c^r)=1 χ ( c r ) = 1 ℓ \ell ℓ [ H P : M ∩ H P ] [H_P:M\cap H_P] [ H P : M ∩ H P ]
N P = ⟨ c r ℓ ⟩ . N_P=\langle c^{r\ell}\rangle. N P = ⟨ c r ℓ ⟩ . The character is surjective modulo ℓ \ell ℓ r r r
Proposition 15.12 (Simultaneous compatible kernels). For the finite orbit list of periodic peripherals and any added maximal cyclic peripherals, all the preceding choices can be made simultaneously and equivariantly. The resulting N D ⊴ D N_D \trianglelefteq D N D ⊴ D S D S_D S D
Proof. Make every finite requirement at the H 0 H_0 H 0 H H H ℓ \ell ℓ F \mathbb{F} F
Every final kernel lies in the corresponding frozen H H H H H H m Z , n Z m\mathbb{Z}, n\mathbb{Z} m Z , n Z k Z k\mathbb{Z} k Z lcm ( gcd ( m , k ) , gcd ( n , k ) ) \operatorname{lcm}(\operatorname{gcd}(m,k),\operatorname{gcd}(n,k)) lcm ( gcd ( m , k ) , gcd ( n , k )) gcd ( lcm ( m , n ) , k ) \operatorname{gcd}(\operatorname{lcm}(m,n),k) gcd ( lcm ( m , n ) , k )
Filled vertex groups and central panels
Fix a terminal folded vertex row J = Y v J=Y_v J = Y v J ‾ \overline{J} J J J J P P P P d = S d P_d=S_d P d = S d
Here is the local construction we seek. Free-group Cayley trees supply horizontal edges, whose original vertices are called ordinary sites. A vertex representing a peripheral coset is called an apex. After filling, its central incidences form a finite link graph: link vertices index radial edges to ordinary sites, and link edges record horizontal edges between those sites. Binary labels on the link vertices prescribe radial cochain values. Where the labels change, the intervening horizontal edge must have mod-two cochain total one. We match these cut marks at copies of shared horizontal midpoints. Large angles between the corresponding radial edges will make the matched components quasiconvex trees. We first identify the central incidences and their cyclic attachments, then build the fan model in which this matching takes place. The exact covers from the preceding section supply the binary labels; the next section extends their cochains across the cyclic attachments by cancelling odd boundary totals.
The central components before filling
Lemma 16.1 (The component presentation). The developed presentation of J J J (i) A central component has a bipartite tree of groups, with finite-rank free groups F F F P d P_d P d M ≤ P d M \leq P_d M ≤ P d F F F
(ii) Each edge between central components has infinite cyclic group ⟨ c ⟩ \langle c\rangle ⟨ c ⟩ d d d d d d
(iii) The unused relative occurrences are cyclic. We call them outer occurrences. Internal and outer nodes together label the actual peripheral cosets, without duplication. The developed data have finite quotient by J J J
Every included S d S_d S d
Proof. There are three cases.
For a free actual row X x X_x X x Γ v \Gamma_v Γ v S d S_d S d M x d M_x^d M x d
Suppose next that the actual row has a nonfree root, and choose one of its slots. Write the root splitting, with all translates understood, as
E ∗ ⟨ z ⟩ ⟨ u a ⟩ ∪ U , U = ⟨ u ⟩ , E *_{\langle z\rangle} \langle u^a\rangle\cup U,\qquad U=\langle u\rangle, E ∗ ⟨ z ⟩ ⟨ u a ⟩ ∪ U , U = ⟨ u ⟩ , where E E E F ~ = X x ∩ Y f = X x \widetilde{F}=X_x\cap Y_f=X_x F = X x ∩ Y f = X x x x x f f f d d d K f K_f K f K f ⊂ Γ v K_f\subset\Gamma_v K f ⊂ Γ v ( x , d ) (x,d) ( x , d ) M x d M_x^d M x d E E E K f K_f K f Γ f \Gamma_f Γ f Z d Z_d Z d
A missing incidence therefore has cyclic group M x d M_x^d M x d E E E K f K_f K f Y f Y_f Y f Γ f \Gamma_f Γ f E e E_e E e P d = S d P_d=S_d P d = S d M e d M_e^d M e d P d P_d P d X x ≤ Y f X_x\leq Y_f X x ≤ Y f d d d
Finally suppose that root powers are unclaimed. The additional maximal cyclic family U ∗ U^* U ∗ M x d M_x^d M x d H 1 ( M x d ) H_1(M_x^d) H 1 ( M x d ) M x d M_x^d M x d M x d M_x^d M x d rk ( M x d ) − 1 \operatorname{rk}(M_x^d)-1 rk ( M x d ) − 1 M x d M_x^d M x d
Cut Γ v \Gamma_v Γ v S d S_d S d U U U U U U U ∗ U^* U ∗ J J J U U U x x x Γ v \Gamma_v Γ v X x X_x X x x x x U U U ⟨ u a ⟩ \langle u^a\rangle ⟨ u a ⟩ S d S_d S d S d S_d S d X x X_x X x
In all three cases the preceding cocompactness results give finite quotient data. Positive period dimension at a free level supplies a central incidence at every included S d S_d S d
A tagged fan complex
Build a J J J X \mathcal{X} X
At a free node use its Cayley tree for a finite free basis. Subdivide each positive oriented edge by a midpoint. The original vertices are ordinary sites ; the two subdivided edges are horizontal half-edges . For an incidence group M M M [62 ] , Sections 3, 5 and 7. Cone this subtree from its internal apex d d d
For a component edge with group ⟨ c ⟩ ≤ P d \langle c\rangle\leq P_d ⟨ c ⟩ ≤ P d c c c d d d satellite arm . Choices are made on representatives of developed component-edge orbits and then extended equivariantly. The cyclic group acts on the indexed line by translation; consequently no further compatibility inside the target X x X_x X x
For an outer occurrence p p p P p = ⟨ c p ⟩ P_p=\langle c_p\rangle P p = ⟨ c p ⟩ c p c_p c p p p p
Each central component is simply connected: it is a tree of simply connected free-node spaces, point spaces, and incidence subtrees. The satellite construction joins these components along their component tree. Each joining slab is the mapping cylinder of its indexed line’s boundary map, with the other end collapsed to its source point. Van Kampen’s theorem therefore preserves simple connectivity. Coning the remaining outer indexed lines preserves it as well.
There are finitely many cell types modulo J J J J J J g ∈ J g \in J g ∈ J
g t or g t P (45) gt \quad\text{or}\quad gtP \tag*{(45)} g t or g tP (45) for fixed offsets t t t Figure 4 .
Figure 4. The three fan types before filling. The two d d d d d d p p p
Let K J K_J K J J J J
X ‾ = K J \ X , A = X ‾ ( 1 ) . \overline{X}=K_J\backslash X,\qquad\mathcal{A}=\overline{X}^{(1)}. X = K J \ X , A = X ( 1 ) . Lemma 16.2 (The filled model). For sufficiently deep choices, X ‾ \overline{X} X A \mathcal{A} A J ‾ = J / K J \overline{J}=J/K_J J = J / K J P ‾ = P / N P \overline{P}=P/N_P P = P / N P A \mathcal{A} A J ‾ \overline{J} J
The central link L N ( d ) L_{N}(d) L N ( d ) N P N_P N P M ∩ N P M\cap N_P M ∩ N P n = ∣ c ˉ ∣ n=|\bar{c}| n = ∣ c ˉ ∣
Proof. The kernel K J K_J K J X X X K J K_J K J X X X
Exact peripheral reduction gives K J ∩ P = N P K_J\cap P=N_P K J ∩ P = N P N P N_P N P J ‾ \overline{J} J ∣ c ˉ ∣ |\bar{c}| ∣ c ˉ ∣
Formula (45) maps every edge to a bounded path in the coned Cayley graph of the induced peripheral structure. Conversely, connectivity of X \mathcal{X} X A \mathcal{A} A J ˉ \bar{J} J ˉ
For germs e , e ′ e,e' e , e ′ s s s ∠ s ( e , e ′ ) \angle_s(e,e') ∠ s ( e , e ′ ) A ∖ { s } \mathcal{A}\setminus\{s\} A ∖ { s } ∞ \infty ∞
Lemma 16.3 (The finite angle dictionary). All of the following conditions can be imposed for any finite list of bounds before choosing the filling.
(i) There are no self-spokes. Two distinct central spokes from a fixed apex do not end at the same physical horizontal midpoint, and likewise do not end at the same ordinary site.
(ii) For each R R R R R R
(iii) On one indexed arm, source germs at bounded angle have sites whose cyclic period coordinates differ by a uniformly bounded amount.
(iv) Every bounded closed walk in which each apex is visited at most once cyclically has a closed labeled lift to the original fan graph, with its given edge types and projected incidences.
(v) All arm orders used here can be chosen even and arbitrarily large.
Proof. The first assertion holds upstairs. Satellite targets belong to components different from their source component. Within the central bipartite tree, at most one incidence joins a specified pair of nodes, and its incidence subtree is embedded in the free tree. A violation after filling would give an equality of endpoint frames, or membership of a fixed offset in a specified peripheral coset. There are finitely many edge types, so the finite-element and coset-avoidance tests from Section 14 exclude these violations.
For the second assertion, expand an avoiding path in A \mathcal{A} A (45) . Add its finitely many offsets to the ordinary generating letters before setting the depth. Each expanded edge uses ordinary offsets and, when necessary, the cone legs at its endpoints. Neither endpoint is s s s s s s [53 , 55 ] . Tags and their free edge frames bound the number of germs represented by that list.
For two sites on the same arm, normalize both word phases and write the relative rotation as a power of its generator c c c ⟨ c ⟩ \langle c\rangle ⟨ c ⟩
Finite-list membership avoidance inside the ambient hyperbolic VCS periodic peripheral ensures that such membership already holds before filling. Since N P N_P N P ⟨ c ⟩ \langle c\rangle ⟨ c ⟩
For the fourth assertion, the rest of the closed walk is an avoiding path at each of its apex visits. Replace the successive normalized peripheral transporters by elements of the finite lists in the second assertion. At regular vertices the transitions are fixed offsets. After normalizing an initial oriented edge, the closing equation is one of finitely many actual word equations. Injection on that finite set makes it hold upstairs. This constructs one closed labeled lift; it does not assert compatible lifts of arbitrarily many independent translates. That stronger assertion will not be needed.
Finally use the divisible order choices and exact order conclusions of Proposition 15.10. Only finitely many bounds and element lists have been imposed, so all the choices are simultaneous.
Lemma 16.4 (Exact graph-cover data ). Let H P H_P H P periodic peripheral 's normal base choice H H H P d P_d P d L H ( d ) L_H(d) L H ( d ) H P H_P H P B B B ℓ \ell ℓ B j → B B_j \to B B j → B ℓ > 1 \ell> 1 ℓ > 1 F \mathbb{F} F char F ∤ ℓ \operatorname{char}\mathbb{F} \nmid\ell char F ∤ ℓ L N ( d ) L_N(d) L N ( d )
(i) the kernels of all these cyclic covering maps; and
(ii) inside each covering kernel, the kernels of all additive first cohomology classes in the nontrivial deck part of H 1 ( B j ; F ) H^1(B_j;\mathbb{F}) H 1 ( B j ; F )
The assertion holds also for the cyclic root incidences, whose nontrivial deck cohomology part is zero.
Proof. Apply Corollary 15.9 to the actual free-row incidence M M M M x d M_x^d M x d ℓ \ell ℓ
Large-angle comparisons
We record the geometry used at each later wall gluing. All constants in this subsection depend on uniform hyperbolicity and the displayed quasigeodesic constants, not on the final peripheral orders.
Lemma 16.5 (Forced passage and marker comparison ). In a uniformly hyperbolic graph the following hold.
(i) Geodesic legs from s s s s s s
(ii) *Two uniform quasigeodesics from s s s s s s s s s s s s s s s
(iii) Suppose uniform quasi-isometrically embedded tree pieces are joined by switches whose spokes have uniformly bounded lengths. A switch maps to a marker in the graph. For every reduced incidence path, assume each connector through a piece has distinct endpoint markers and avoids them except at its endpoints. Impose the same conditions on partial connectors at marked-node endpoints. Assume also that each piece attaches at most once at a given marker, and that all distinct spoke germs at each switch have pairwise angle larger than a sufficiently large fixed threshold. Then the joined components are uniformly quasi-isometrically embedded trees. Their node projections are injective provided the piece node projections are injective and the stated endpoint conditions hold. Every interior switch on a reduced path is a large-angle turn on a geodesic between its projected endpoints. The same conclusion holds towards an ideal endpoint of a ray when the ambient graph is proper.
The quasi-isometry constants in the third assertion can be fixed after one minimum angle threshold is met. Increasing the angle threshold later increases the forced comparison angle without changing those constants.
Proof. For the first two assertions use thin triangles and quasigeodesic stability. If a comparison path avoids s s s a a a s s s s s s s s s
For the gluing assertion, every piece connector is a uniform quasigeodesic: adding bounded end spokes to a uniformly embedded piece path changes only its additive constants. The endpoint avoidance hypotheses allow its departure and arrival germs to be compared with those of a geodesic replacement. The angle error is bounded by the second assertion. Hence consecutive replacement geodesics still make an arbitrarily prescribed large angle at their common switch if the spoke threshold is chosen sufficiently large.
Apply the first assertion inductively along a reduced incidence path. The concatenation of its geodesic replacements is a geodesic, and every interior switch remains a large-angle vertex on it. A circuit not contained in a piece would therefore give a positive-length geodesic returning to its start; a circuit within a piece is impossible because the piece is a tree. Reusing one attachment does not create an exception, since the incidence path was reduced and each piece has at most one attachment at that marker. The same argument rules out repetition of node projections across switches; within pieces use the assumed injectivity.
There are uniform constants a , b a,b a , b a a a b b b
Sparse cuts from exact homological covers
A central panel b b b d d d ζ b : V ( L N ( d ) ) → { 0 , 1 } \zeta_b : V(L_N(d)) \to\{0,1\} ζ b : V ( L N ( d )) → { 0 , 1 } cuts are link edges with unequal endpoint labels. The corresponding cut germs are the central spokes to their midpoints.
Proposition 16.6 (Central panel tests). Fix K K K q q q r r r B B B L N L_N L N B B B K K K K K K
There are also fillers: for every occurrence of a link edge there is a panel cutting that occurrence whose cuts all lie over its one base edge in L H L_H L H K K K
Proof. We first arrange that different lifts of any one edge of L H L_H L H K K K H P H_P H P H P / N P H_P/N_P H P / N P H H H N P ≤ H P N_P \leq H_P N P ≤ H P H P H_P H P H H H
Select support base edges successively. At each stage forbid all base edges having a lift angularly near a prescribed direction or near any lift of a support edge already chosen. The number forbidden by a direction is uniformly bounded. The number forbidden by one support edge is also uniformly bounded: the deck group H P / N P H_P/N_P H P / N P ( q 2 ) \binom{q}{2} ( 2 q ) B B B B B B L H L_H L H B B B
Here is the separation assertion underlying the construction. Let u , v u,v u , v L N L_N L N F F F d L N ( u , v ) > B d_{L_N}(u,v)>B d L N ( u , v ) > B B j B_j B j c j c_j c j
A j = 1 ℓ ∑ g ∈ Deck ( B j / B ) g A_j=\frac{1}{\ell}\sum_{g\in\operatorname{Deck}(B_j/B)}g A j = ℓ 1 g ∈ Deck ( B j / B ) ∑ g for deck averaging on chains and cochains. For each allowed base edge, differences of the cochains on two of its individual oriented lifts have coefficients 1 , − 1 1,-1 1 , − 1 ker A j \ker A_j ker A j L N L_N L N T \mathbb{T} T
Suppose none of these potentials distinguishes u , v u,v u , v c j c_j c j
( 1 − A j ) c j is supported on the inverse image of F , ∂ ( 1 − A j ) c j = ( 1 − A j ) ( [ v j ] − [ u j ] ) . (46) (1-A_j)c_j\text{ is supported on the inverse image of }F,\qquad\partial(1-A_j)c_j=(1-A_j)([v_j]-[u_j]). \tag*{(46)} ( 1 − A j ) c j is supported on the inverse image of F , ∂ ( 1 − A j ) c j = ( 1 − A j ) ([ v j ] − [ u j ]) . (46) Each lift of a base tree of F F F ℓ \ell ℓ u , v u,v u , v (46) would have nonzero augmentation on at least one lift of either tree: its endpoint mass is the difference between one point and its deck average, and ℓ > 1 \ell> 1 ℓ > 1 ℓ \ell ℓ T \mathbb{T} T u j , v j u_j, v_j u j , v j
Let f f f f j f_j f j u j u_j u j v j v_j v j B j B_j B j ( 1 − A j ) c j (1-A_j)c_j ( 1 − A j ) c j ( 1 − A j ) f j (1-A_j)f_j ( 1 − A j ) f j c j − f j c_j-f_j c j − f j f f f u , v u,v u , v L N L_N L N d L N ( u , v ) ≤ ∣ F ∣ ≤ B d_{L_N}(u,v) \leq|F| \leq B d L N ( u , v ) ≤ ∣ F ∣ ≤ B
Now process all same-component target pairs at distance greater than B B B
For a filler, choose a cyclic cover over the component in question and take the difference cochain between the lift containing the specified occurrence and a different lift of its base edge. Its pullback is exact and its value on the specified edge is nonzero. A binary separation of the two endpoint potential values therefore cuts that occurrence, with all cuts over that one base edge. This argument only requires the nontrivial covering degree. In particular, when the nontrivial cohomology itself is zero, the difference cochain is already exact but still has a nonconstant potential. The proof works unchanged in characteristic two when ℓ \ell ℓ
Central trees and their cochains
For the actual finite links, take finite systems realizing all required panel tests and then all peripheral translates. Decorate panels by free regular P ‾ d \overline{P}_d P d
For each central spoke orbit type, a peripheral orbit of fillers adds a positive constant number of cuts at every spoke of that type and none of the other types. Multiply the requested systems by a common multiple of these positive numbers. Choose a common larger integer ℓ \ell ℓ ℓ \ell ℓ ℓ \ell ℓ ℓ \ell ℓ
Lemma 16.7 (Central trees). Every component W 0 W_0 W 0 A \mathcal{A} A larger thresholds. There are finitely many decorated node and edge orbits. Distinct systems may be retained separately as a finite labeled pool.
Proof. Apply Lemma 16.5 with slots as point pieces and panels as switches. Along a reduced path the two panel markers adjacent to one slot are distinct: from a fixed apex, that slot is matched to only one cutting panel. The two-spoke connector through the slot avoids its endpoint apices, and each point piece attaches at most once over any apex. At a panel all cut germs have the stipulated large angle. The marker lemma proves both the tree assertion and injectivity on nodes. Finiteness of the orbit data follows from finite links, finite peripheral reductions, and the finite multiplicities just chosen.
A horizontal edge is marked by W 0 W_0 W 0 W 0 W_0 W 0 W 0 W_0 W 0 W 0 W_0 W 0
Lemma 16.8 (The central cochain). Each W 0 W_0 W 0 τ 0 W 0 \tau_0^{W_0} τ 0 W 0 ζ b ( t ) \zeta_b(t) ζ b ( t ) d − t d-t d − t b b b W 0 W_0 W 0 W 0 W_0 W 0
Proof. The assertions specify the ordinary radial and horizontal values. For an edge with positive orientation u − m − v u-m-v u − m − v ϵ \epsilon ϵ ζ b ( u ) + ϵ = ζ b ( v ) \zeta_b(u)+\epsilon=\zeta_b(v) ζ b ( u ) + ϵ = ζ b ( v ) b b b
Lemma 16.9 (Satellite arm exclusivity and bounded support). If W 0 W_0 W 0 d d d d d d d d d τ 0 W 0 \tau_0^{W_0} τ 0 W 0
If τ 0 W 0 \tau_0^{W_0} τ 0 W 0 P ˉ d \bar{P}_d P ˉ d
Proof. An anchor on the arm is joined to d d d W 0 W_0 W 0 d d d d d d W 0 W_0 W 0 d d d W 0 W_0 W 0
The lift of the W 0 W_0 W 0 d d d P d / ⟨ c ⟩ P_d/\langle c\rangle P d / ⟨ c ⟩ P ˉ d \bar{P}_d P ˉ d
The path in W 0 W_0 W 0 d d d d d d P ‾ d \overline{P}_d P d W 0 W_0 W 0
Odd seeds on satellite arms
Proposition 16.10 (Odd inner seeds). A finite pool of central panel systems, chosen at successive fixed angle scales, can be required to contain, on every satellite arm, a central tree W 0 W_0 W 0 τ T 0 W 0 \tau_{T_0}^{W_0} τ T 0 W 0
Proof. Fix an arm cycle β \beta β d d d n n n s s s d d d s s s d d d s s s d d d
Use Proposition 16.6 with cuts sufficiently far from these directions and with every binary assignment on its bounded diameter target classes. For the tree through such a testing panel at s s s w ≠ s w \ne s w = s β \beta β d d d s s s w w w s s s d d d w w w s s s β \beta β β \beta β s s s
If any test has odd total, it supplies the seed. Otherwise every target class occurs an even number of times. Pair the radial half-incidences within each class and replace each pair, as a mod-two chain, by a horizontal link path between its two ordinary sites. The paths have a uniform length bound B 1 B_1 B 1 z z z
We justify that z z z C n = R / n Z C_n=\mathbb{R}/n\mathbb{Z} C n = R / n Z β \beta β B 1 B_1 B 1 d d d d d d s s s
For completeness, let T T T D ( R ) D(R) D ( R ) R R R d d d
M = max { 1 , D ( T ) , D ( 2 T + 1 ) } , n > 8 M . M=\max\{1,D(T),D(2T+1)\},\qquad n>8M. M = max { 1 , D ( T ) , D ( 2 T + 1 )} , n > 8 M . For every physical vertex in the finite graph used by β \beta β T T T M M M [ − M , M ] [-M,M] [ − M , M ] 1 + 2 M < n / 2 1+2M<n/2 1 + 2 M < n /2 n n n s s s s s s 2 T + 1 2T+1 2 T + 1 2 M < n / 2 2M<n/2 2 M < n /2 z z z z z z
Choose a second central panel system with cut angles larger than a threshold fixed after B 1 B_1 B 1 d d d d d d d d d
Nor can this tree contain two distinct horizontal anchors of z z z β \beta β d d d d d d d d d z z z z z z β \beta β
There are only finitely many actual links and bounded-size requests for a fixed filling. Include all first-stage tests, their translates, and the second-stage filler systems. The constants determining these requests were fixed independently of n n n
We make explicit the quantifier order used here and in the next section. First fix the original row types, their word paths, the uniform relative hyperbolicity bounds, and a minimum bend threshold in Lemma 16.5. This fixes the quasi-isometry constants of the central trees and the arm-support bounds in Lemma 16.9. Any larger cut separations preserve these constants.
Next prescribe the finitely many target-list sizes and direction avoidance thresholds that will be tested. Proposition 16.6 provides a bound for the number of forbidden base edges and for the resulting class diameters, independently of H H H N P N_P N P B 1 B_1 B 1
Finally choose the normal base subgroups H H H N P ≤ H P N_P \le H_P N P ≤ H P
The outgoing data are therefore a finite pool of equivariant central trees W 0 W_0 W 0 τ 0 \tau_0 τ 0
Switches, cocycles, and boundary separation
Our goal is to separate every pair of boundary points by a quasiconvex subgroup, so that the boundary criterion gives a cocompact cubulation of each filled row. We first extend the central trees and cocycles across the satellite fans and then the outer fans. The second half of the section proves that the resulting cocycles supply all the required separators.
We continue with a terminal vertex row J = Y v J = Y_v J = Y v A \mathcal{A} A J ˉ \bar{J} J ˉ A \mathcal{A} A d d d p p p
All chains and cochains in this section have coefficients in F 2 \mathbb{F}_2 F 2 pure internal path avoids outer apices and uses true ordinary neighbors at every internal apex and horizontal midpoint. The boundary word of every satellite or outer fan has this property. Pair replacements will produce chains, and these need not be connected.
We use the central trees W 0 W_0 W 0 τ 0 \tau_0 τ 0 anchor is a projected panel or slot node of such a tree, and later also a switch node. In particular, a central radial contribution has a panel anchor at its apex, whereas a horizontal contribution has a slot anchor at its midpoint. The angle dictionary of Lemma 16.3 and the marker comparison of Lemma 16.5 will always be used with bounds fixed before the final filling kernels are chosen. We give the full order of these choices at the end of the section.
Nonmixed panels and the first switches
At an internal apex d d d d d d
Definition 17.1. A side is a pair ( W 0 , a ) (W_0,a) ( W 0 , a ) W 0 W_0 W 0 a a a d d d odd if τ 0 W 0 \tau_0^{W_0} τ 0 W 0 a a a a a a
A central-only total panel is a central panel b b b d d d satellite-only total panel T T T W 0 W_0 W 0 d d d T T T d d d T T T
On each satellite arm, the binary labels of T T T T T T
Figure 5 shows these two kinds of abstract panel over one physical apex. The fan spaces themselves were shown in Figure 4 .
Figure 5. Nonmixed panels in the filled complex. The central-only panel b b b T T T d d d ( W 0 , a ) (W_0,a) ( W 0 , a ) W 0 W_0 W 0 d d d b b b
Lemma 17.2 (Satellite profile tests). Fix bounds on the number of target ordinary satellite sites and forbidden germ directions at d d d
Proof. On each arm, order the listed sites cyclically. Join consecutive listed sites into one cluster when the intervening gap is smaller than a fixed threshold D D D D D D
The seed’s support has uniformly bounded width. Its chosen anchor rotates freely, and the angle dictionary bounds the number of rotations whose spoke lies angularly close to a forbidden direction or a previously placed spoke. There are only boundedly many directions and earlier placements. Choose D D D d d d
Lemma 17.3 (Equivariant consumption). After taking finitely many labeled copies of a finite pool of central systems, every odd side can be consumed exactly once. This can be done equivariantly, with finitely many decorated panel and edge orbits, while including all the tests of Lemma 17.2 and any fixed lower bound on switch angles.
Proof. For each of the finitely many apex types, include the full free P ‾ d \overline{P}_d P d m m m
Consider an arm with even period n n n c c c c n / 2 c^{n/2} c n /2 P ‾ d \overline{P}_d P d n n n m m m
Consume the unused copies by antipodal pair panels. The pair’s germs have arbitrarily large angle once the period-coordinate avoidance test is imposed. An unordered pair may have an order-two stabilizer, but a free orbit of decorated pair panels uses two copies of each side in such a pair. The even residual multiplicities therefore permit these free orbits; the offsets of their binary labels are transported with the decorations. There is no need to choose an offset invariant under an involution that exchanges the two sides. More explicitly, write a = c n / 2 a=c^{n/2} a = c n /2 i , j i,j i , j g g g ( g s , i ) (gs,i) ( g s , i ) ( g a s , j ) (gas,j) ( g a s , j ) g a ga g a ( g a s , i ) (gas,i) ( g a s , i ) ( g s , j ) (gs,j) ( g s , j )
The stabilizer of an odd side at d d d d d d W 0 W_0 W 0
Assignments at distinct source apices impose separate incidence conditions. Each labeled W 0 W_0 W 0 d d d
Different central systems in the pool retain their tags, slots, and cochains. Taking their disjoint union does not require rematching slot data. If satellite tests are not requested in a particular system, the antipodal construction alone suffices, without an odd-seed hypothesis.
Let W 1 W_1 W 1 W 0 W_0 W 0
Lemma 17.4 (The internal trees and their cocycles). The W 1 W_1 W 1 τ 1 \tau_1 τ 1 s → t s \to t s → t
(i) W 1 W_1 W 1 s s s
(ii) a constituent W 0 W_0 W 0 t t t t t t
The bounds, and the quasi-isometry constants, depend only on minimum bend requirements and the basic central support bounds. Increasing subsequent angular separations does not increase these constants.
Proof. Use the satellite-only panels as markers in Lemma 16.5. By arm exclusivity, a piece attached at d d d d d d W 1 W_1 W 1
On the central graph define τ 1 \tau_1 τ 1 τ 0 \tau_0 τ 0 a a a d d d T T T d d d even encounter of a constituent W 0 W_0 W 0 a a a n n n
Every odd encounter is already accounted for: its consuming panel belongs to the same component. A side consumed on a different arm at d d d a a a τ 0 \tau_0 τ 0
A default can be nonzero only within the bounded enlargement of its support cluster, proving (ii). The direct panel labels give (i), regardless of their additive constants. The same reasoning handles nondinary boundary sites. For a fixed edge there are only finitely many possible nearby anchors, by local finiteness of the filled graph and the finite decorated slots and links. Node uniqueness prevents infinitely many constituent pieces from using a fixed such anchor. All sums are thus defined edge by edge. Every normalization and decoration is transported equivariantly.
Outer switches and boundary sides
An outer encounter is similarly a pair ( W 1 , p ) (W_1,p) ( W 1 , p ) p p p τ 1 \tau_1 τ 1 W 1 W_1 W 1
Lemma 17.5 (Outer support clusters). For a fixed W 1 W_1 W 1 τ 1 \tau_1 τ 1 p p p
Proof. By Lemma 17.4, each nonzero position has an anchor at uniformly bounded internal distance. Any two such anchors are at bounded ambient distance by the shortcut through p p p W 1 W_1 W 1 p p p p p p
At each outer apex, pair every odd W 1 W_1 W 1 p p p W 1 W_1 W 1 p p p
Proposition 17.6 (Completed trees and cocycles). The resulting components W = W 2 W = W_2 W = W 2 W 1 W_1 W 1 τ W \tau_W τ W W W W
Proof. The period-coordinate dictionary makes the two germs of every outer switch as far apart as any prescribed fixed threshold. The W 1 W_1 W 1 W 1 W_1 W 1
On the internal graph, sum the cocycles of the constituent W 1 W_1 W 1
The difference of successive radial labels is the internal boundary cochain. Thus every outer triangle has cocycle sum zero. Together with Lemma 17.4, this proves closedness on every face. Even defaults are supported near anchors; the paired contribution, even when its primitive is nonzero along a long interval, lies on edges incident to its switch p p p W W W
Changing the constant for the paired primitive changes precisely the values of all radial edges incident to that outer apex. Since that apex has just its one fan arm, the change is the coboundary of its vertex indicator. The possible order-two stabilizer of the unordered pair therefore creates exactly this permitted ambiguity and no further one. All even normalizations are equivariant.
Finally, the central data, finite links, decorated profiles, and labeled copies have finite orbit sets. At an outer cycle only finitely many nearby anchors contribute, so the pair-switch and chosen spoke data also have finitely many orbits. This proves the last assertion.
Lemma 17.7 (From a cocycle to boundary separation). *For each W W W J ˉ \bar{J} J ˉ W W W ∂ J ‾ ∖ Λ W \partial\overline{J} \setminus\Lambda W ∂ J ∖ Λ W Λ W \Lambda W Λ W
Proof. If two decorated nodes of one component are in the same ambient group orbit, any group element carrying one to the other preserves their component. Finiteness of the ambient decorated orbit sets therefore gives a finite quotient of W W W A \mathcal{A} A Λ W \Lambda W Λ W
Simple connectivity lets us integrate τ W \tau_W τ W Λ W \Lambda W Λ W W W W W W W W W W
Under an element preserving W W W Λ W \Lambda W Λ W H H H τ W = δ f \tau_W=\delta f τ W = δ f g g g W W W g τ W + τ W = δ η g g\tau_W+\tau_W=\delta\eta_g g τ W + τ W = δ η g η g \eta_g η g
g f + f + η g = ϵ ( g ) ( ϵ ( g ) ∈ F 2 ) . gf+f+\eta_g=\epsilon(g) \qquad(\epsilon(g)\in\mathbb{F}_2). g f + f + η g = ϵ ( g ) ( ϵ ( g ) ∈ F 2 ) . Off the invariant switch set this is g f + f = ϵ ( g ) gf+f=\epsilon(g) g f + f = ϵ ( g ) ϵ \epsilon ϵ η g \eta_g η g
Choose a fixed neighborhood of an H H H H H H ∂ J ‾ ∖ ∂ H \partial\overline{J}\setminus\partial H ∂ J ∖ ∂ H [7 ] , Theorem 1.4, including the case of a finite subgroup separating an already disconnected boundary.
Comparison paths and satellite tests
We now prove that finitely many successive systems provide the separators. There are three geometric cases: a large outer gap; a large internal turn after bounded outer turns have been replaced; and the remaining case in which all turns are bounded. In parallel, a comparison around an outer boundary cycle constructs the odd W 1 W_1 W 1
The two uses have different outputs. A circle comparison starts with one outer boundary cycle and produces an odd W 1 W_1 W 1 W 2 W_2 W 2 W 1 W_1 W 1 A p A_p A p
Lemma 17.8 (Replacing bounded outer turns). Fix a bound A p A_p A p on cyclic period gaps at outer turns of a bi-infinite geodesic γ \gamma γ Replacing these turns by boundary arcs and making the elementary fan detours produces a pure internal uniform quasigeodesic α \alpha α Its constants depend only on A p A_p A p and the fixed data.
(i) Each physical vertex occurs a bounded number of times on α \alpha α and all its occurrences lie in a bounded parameter interval.
(ii) If an internal apex d d d lies on γ \gamma γ all new germs at d d d lie within bounded angular distance of one of its two original germs ν − , ν + \nu_-,\nu_+ ν − , ν + When these are sufficiently far apart, each neighborhood contains an odd number of the new radial half-incidences.
(iii) If all original internal turns have bounded angle, then the angular diameter of all germs at all visits to any one internal apex on α \alpha α is bounded.
Proof. A period gap of at most A p A_p A p
An elementary radial detour changes its germ at d d d d d d d d d
For (iii), a new internal passage in an inserted arc has a short avoiding comparison through the parent outer apex. All individual turns are therefore bounded. Between consecutive visits to an internal apex, the subpath avoids that apex and has bounded length by (i). It bounds the angle between the corresponding visit germs, proving the assertion about cumulative diameter. Arguments excluding outer switches require only avoidance of outer apices, and do not need (iii).
There will be two comparison settings. In the line setting , α \alpha α W 2 W_2 W 2 circle setting , α \alpha α p p p W 1 W_1 W 1 p p p p p p
When every test is even, we will pair radial occurrences and replace each pair by a short boundary path. The replacement is a sum of fan-triangle boundaries, so it preserves cocycle evaluation. It may disconnect the chain. The invariant that keeps the final horizontal chain nonzero is mod-two winding in the circle setting, and mod-two crossing from one end to the other in the line setting. We verify these invariants in Lemma 17.13.
Lemma 17.9 (Satellite tests and uncontaminated evaluations). At an internal apex d d d visited by α \alpha α apply Lemma 17.2 to its satellite radial site list, with spoke germs far from every d d d direction of α \alpha α The resulting tree through T T T evaluates on α \alpha α exactly as the prescribed satellite labels on that list. In the line setting its limit set avoids both ideal endpoints. If all such evaluations are even, the satellite radial half-incidences can be paired within their classes and replaced by bounded purely central boundary arcs avoiding d d d
Proof. First consider comparisons from d d d W 1 W_1 W 1 W 1 W_1 W 1 T T T
If another anchor w ≠ d w \ne d w = d α \alpha α d d d w w w d d d d d d p p p d d d
It remains to check defaults rather than only exact anchors. A default at a satellite radial site of d d d d d d s ≠ d s \ne d s = d W 0 W_0 W 0 w w w w ≠ d w \ne d w = d α \alpha α s s s w w w d d d w = d w=d w = d W 0 W_0 W 0 d d d T T T W 1 W_1 W 1 W 2 W_2 W 2 s ≠ d s \ne d s = d
If every class has even occurrence count, pair its half-incidences. Two sites in a class lie on one arm at bounded cyclic distance, so replace the pair of radials by the intervening bounded boundary arc. These satellite boundaries are pure central paths and avoid their source d d d B s B_s B s
For infinite chains this is first a local chain identity. Whenever it is used for a particular separating tree, we have already excluded that tree’s limit set at both ends. Its support then misses the tails and all bounded-parameter replacement patches there, so only finitely many patches affect the evaluation. This justifies the infinite-chain version without an infinite parity sum.
Large internal turns
Lemma 17.10. In the line setting, a sufficiently large original internal turn of γ \gamma γ A d A_d A d B s B_s B s
Proof. Let d d d ν − , ν + \nu_-,\nu_+ ν − , ν + A d A_d A d d d d α ′ \alpha' α ′ d d d A d A_d A d B s B_s B s d d d ν ± \nu_\pm ν ±
Choose a central-only panel b b b ν − \nu_- ν − ν + \nu_+ ν + A d A_d A d d d d
We claim that every first direction in the W 1 W_1 W 1 b b b ν ± \nu_{\pm} ν ± b b b s s s W 0 W_0 W 0 b b b s s s d d d d → s d \to s d → s ν \nu ν w ≠ d w \ne d w = d b b b W 0 W_0 W 0 s s s W 0 W_0 W 0 d d d d d d s s s
Compare from the last d d d α \alpha α W 1 W_1 W 1 W 2 W_2 W 2
We check every possible extra contribution on α ′ \alpha' α ′ w ≠ d w \ne d w = d α \alpha α d d d d d d
A satellite half remaining at s s s s ≠ d s \ne d s = d α \alpha α w w w w ≠ d w \ne d w = d s s s w = d w=d w = d W 0 W_0 W 0 b b b s s s d d d d d d s s s d → s d \to s d → s ν ± \nu_{\pm} ν ± W 0 W_0 W 0 d d d
Only the intended central labels at d d d γ \gamma γ
Regular comparisons and the last horizontal chain
We next treat both the circle setting and the line setting in which every original internal turn is at most A d A_d A d α \alpha α α 1 \alpha^1 α 1
Lemma 17.11 (Regular comparison paths). Sites participating in α 1 \alpha^1 α 1 α \alpha α Proof. For a site on an inserted arc, follow that arc to a paired endpoint. Each internal apex passage on this connector has a short angular bypass through the arm’s source apex, which differs from the apex being traversed. For an untouched half, use that half to an ordinary neighbor when needed. From a tested apex, its connector can start with the particular germ being tested.
Join two such tails by the intervening α \alpha α p p p
Finally, each substitution arc has bounded length and stays close to its assigned old positions. In the line setting quasigeodesicity bounds all possible positions assigned to one physical site or apex. In the circle setting the connector tails avoid p p p p p p
Lemma 17.12 (Central tests on the regular chain). At an internal apex s s s α 1 \alpha^1 α 1 W 1 W_1 W 1 α 2 \alpha^2 α 2
Proof. Use a second central system, with satellite switch bends chosen after the regular comparison bounds of Lemma 17.11. At s s s
Every contribution on the pure central chain requires an actual W 0 W_0 W 0 W 0 W_0 W 0 W 0 W_0 W 0
A tree ray asymptotic to either ideal endpoint is excluded in precisely the same order: it cannot use an outer switch, cannot use a satellite switch, and cannot use a first cut. Thus an odd evaluation gives the claimed separation. In the circle setting it gives the required odd total without any endpoint assertion.
If all tests are even, each bounded intrinsic class has even occurrence count. Pair within these classes and replace the radial pairs by bounded horizontal link paths. Fan triangles preserve every cocycle evaluation. Let B h B_h B h B h B_h B h
Lemma 17.13 (A surviving horizontal edge). *The chain α 2 \alpha^2 α 2
Proof. First choose the final system with its central cut separations, satellite switch bends, and, in the line setting, outer switch bends all above the bounds supplied by the connectors after the horizontal replacements. Two distinct horizontal midpoint anchors of α \alpha α W 0 W_0 W 0 W 0 W_0 W 0
We prove nonzero physical parity without assuming that the modified chain is connected. In the circle setting, assign each participating physical vertex its position on the original parameter circle, using its occurrence or the connector to an original site. All ambiguities have bounded cyclic diameter: the connectors and replacement patches avoid the parent outer apex p p p
In the line setting, assign each participating physical vertex a single real parameter, chosen from its occurrence or connector positions on α \alpha α
At a horizontal midpoint the cycle condition says that its two half-edge counts agree, in both settings; local finiteness justifies this condition in the line setting. Thus some full physical horizontal edge has odd multiplicity. Choose a slot at its midpoint in the final system. On a purely horizontal chain only marked slots contribute. No other midpoint of the chain is marked by this tree, as proved above, and node uniqueness allows just this one slot over the chosen midpoint across all constituent pieces. The evaluation is therefore odd.
All evaluations pass back through the substitutions by triangle consistency. In the line setting the previously proved limit avoidance ensures that the support misses sufficiently distant tails and their bounded-parameter patches. Thus only finitely many substitutions affect this particular evaluation, as required. □
Corollary 17.14 (Odd outer encounters). A finite pool of W 1 W_1 W 1 W 1 W_1 W 1
Proof. Apply the circle setting. An odd satellite test suffices. If none exists, perform the satellite pairings. An odd central test then suffices. If none exists, Lemma 17.13 supplies an odd final slot tree. Only the finitely many successive systems just described are used; the bounds depend on the fixed initial circular comparisons and the preceding pairing bounds. This argument uses W 1 W_1 W 1 W 2 W_2 W 2
Large outer turns and completion of separation
Lemma 17.15. A sufficiently large cyclic period gap at an outer turn of a geodesic γ \gamma γ A p A_p A p
Proof. Let p p p A p A_p A p
Every tree path or ray from the switch leaves in one of these two prescribed directions. It is uniformly quasigeodesic and does not revisit p p p γ \gamma γ
We must also exclude cocycle contributions away from this traversal. A nonzero contribution on either side of γ \gamma γ p p p p p p γ \gamma γ p p p
On the traversal through p p p
Proposition 17.16 (Boundary separation). There are finitely many systems of completed trees such that every pair of distinct points of ∂ J \partial J ∂ J
Proof. Choose a bi-infinite geodesic for the pair. A large outer gap is handled by Lemma 17.15. Otherwise normalize it by Lemma 17.8. A large original internal turn is handled by Lemma 17.10. In the remaining case all original internal turns are bounded. An odd satellite test separates by Lemma 17.9; failing that, an odd central test separates by Lemma 17.12; failing both, Lemma 17.13 supplies the final slot separator. Each lemma proves limit avoidance as well as odd evaluation. The finite choice of systems and its compatibility with the filling depths are verified next.
The order of the finite choices
Lemma 17.17 (No circular choice of thresholds). All systems used above can be chosen from a finite pool by finitely many original depth, angle, girth, period-order, and avoidance tests. These tests can be imposed simultaneously with any initial filling-depth request and the finite-path tests of Section 14.
Proof. For clarity, an angular comparison bound below is a value of the fixed bound functions supplied by the full-graph hyperbolicity and angle dictionary, for already chosen path constants and list sizes. The number of original germ types is finite. Thus any finite collection of such requests is a finite collection of original tests, even though the final links and their periods depend on the eventual filling.
First impose the minimum central and internal switch bounds needed for the marker comparison, the bounded polygon tests, and the inner support clusters. They give uniform quasi-isometry constants for W 0 W_0 W 0 W 1 W_1 W 1 W 2 W_2 W 2
These basic constants determine the outer-gap threshold A p A_p A p α \alpha α A p A_p A p basic first-direction and cluster bounds; it does not use later regular-comparison bend requirements. Include the central odd-seed pool from Proposition 16.10 wherever satellite profile tests will be needed.
Fix the first satellite tests from the bounds for α \alpha α B s B_s B s
Fix the angular neighborhoods and cut-avoidance requests for the large-internal-turn test, using only the preceding constants and B s B_s B s A d A_d A d A d A_d A d
Only now use the absence of turns larger than A d A_d A d B h B_h B h
The extra horizontal tails of length at most B h B_h B h
After all these comparison requests have been recorded, choose outer antipodal switch bends above their maximum wherever needed. This is possible by the period-coordinate tests and does not alter the fixed minimum-threshold quasi-isometry constants. In systems without prescribed satellite labels, consume odd sides by antipodal pairs with the requested bend bound; no new seed or class test is needed.
The finite systems for the circle setting give the odd outer encounters before they are used in the large-outer-turn proof. There is no converse dependence: the circle setting uses only W 1 W_1 W 1
Every placement condition, polygon-lift length, cyclic-index bound, minimum period order, girth requirement, and excluded transporter list is now finite and fixed. The kernel construction of Proposition 15.12, including its graph-cover and order conclusions in Corollary 15.9 and Proposition 15.10, together with Section 16.8, imposes these tests before freezing H , ℓ , T , N H,\ell,\mathbb{T},N H , ℓ , T , N
Good fillings and the primitive-extension conclusion
Theorem 17.18 (Virtually compact special filled vertices). For the terminal row and peripheral data constructed in Sections 13–16, one can choose the height-fiber kernels arbitrarily deep, simultaneously satisfying the finite-path tests, so that every filled vertex group J ˉ \bar{J} J ˉ
Proof. Hyperbolicity and the proper cocompact fan model are supplied by the row filling construction. Elementary filled groups are already virtually compact special. For every other filled row, Lemma 17.17 makes the wall constructions and Proposition 17.16 simultaneous. By Lemma 17.7, each boundary pair is separated by the limit set of a quasiconvex codimension-one subgroup. The boundary criterion [7 ] gives a proper cocompact action on a CAT(0) cube complex. Agol’s theorem [2 ] gives a finite-index subgroup acting specially on that complex. The quotient is compact because the original action was cocompact and the index is finite. Thus the conclusion is virtual compact specialness.
Theorem 17.19 (Arbitrarily deep good quotients). Let G G G
Proof. Impose the given initial request together with the finite-path transfer and the finite tests of Lemma 17.17. The terminal transfer, Theorem 14.9, gives the finite quotient splitting; Proposition 14.11 gives ambient quasiconvexity of its edge groups. Its filled vertex groups are hyperbolic virtually compact special by Theorem 17.18. Deep relative filling in G G G
The hyperbolic quasiconvex hierarchy theorem [66 ] , applied to the transferred finite splitting with these vertex groups, makes the whole quotient virtually compact special. Here the edge groups are quasiconvex in the ambient quotient, as required by the transfer result; this is not an inference from arbitrary embeddings of the vertex groups in right-angled Artin groups. The inherited peripheral and filling conclusions are exactly those of the deep relative filling and standalone row constructions. The initial avoidance request was arbitrary.
Theorem 17.20 (Primitive extensions). Every hyperbolic torsion-free primitive extension group of either of Linton’s two forms specified in the introduction is virtually compact special.
Proof. For such a group the primitive data of Section 5 and the opening reduction establish local indicability, coherence, a finite aspherical two-dimensional one-relator model, and acyclicity of its universal-cover chain complex over the rational Linnell division field. The preceding actual-row, closure, and terminal arguments construct the finite malnormal quasiconvex virtually compact special peripheral family. Theorem 17.19 supplies arbitrarily deep good quotients with hyperbolic virtually compact special peripheral images and the inherited filling conclusions. Thus all hypotheses of the filling criterion, Theorem 3.1, hold. That criterion gives virtual compact specialness of the original group G G G
Consequences
Let G G G H ≤ G H \leq G H ≤ G [52 ] first puts it in the first class. We explain the finite-quotient and subgroup consequences before the more specific virtual-fibering result.
Linearity and finite quotients
The finite Salvetti model embeds H H H Z \mathbb{Z} Z [28 ] . Inducing a faithful integral representation from the finite-index subgroup H H H G G G GL N ( Z ) \operatorname{GL}_{N}(\mathbb{Z}) GL N ( Z ) N N N G G G Z \mathbb{Z} Z [10 ] ; the compact-special result here also supplies integral linearity. For finite-rank free bases, this answers Druţu–Sapir’s question about the existence of hyperbolic nonlinear ascending HNN extensions in the negative [20 ] .
Finite quotients also distinguish conjugacy classes. A group is conjugacy separable if any two nonconjugate elements have nonconjugate images in some finite quotient, and hereditarily conjugacy separable if every finite-index subgroup has this property. Minasyan–Zalesskii’s theorem applies to every hyperbolic virtually compact special group, so G G G [49 ] . For one-relator groups with torsion, they had already established this conclusion [48 ] .
There is also a cohomological consequence of the finite special model. Canonical completion realizes H H H [49 ] therefore makes G G G cohomologically good : for every finite G G G M M M n ≥ 0 n \geq0 n ≥ 0
H c o n t n ( G ^ ; M ) ⟶ H n ( G ; M ) H_{\mathrm{cont}}^{n}(\widehat{G};M) \longrightarrow H^{n}(G;M) H cont n ( G ; M ) ⟶ H n ( G ; M ) is an isomorphism. Here G ^ \widehat{G} G
Quasiconvex subgroups and largeness
If Q ≤ G Q \leq G Q ≤ G Q 0 = Q ∩ H Q_{0}=Q \cap H Q 0 = Q ∩ H H H H H 0 ≤ H H_{0} \leq H H 0 ≤ H Q 0 Q_{0} Q 0 H 0 → Q 0 H_{0} \to Q_{0} H 0 → Q 0 [28 ] . In particular, Q 0 Q_{0} Q 0 H H H H ∖ Q 0 H \setminus Q_{0} H ∖ Q 0 Q 0 Q_{0} Q 0 Q Q Q G G G [28 ] . This permits finite covers that retain a specified quasiconvex subgroup while excluding any prescribed finite set outside it. The retraction gives the stronger structural control on Q ∩ H Q \cap H Q ∩ H
When G G G non-elementary , meaning that it is not virtually cyclic, it is also large : some finite-index subgroup surjects onto a free group of rank two. Indeed, the torsion-free hyperbolic subgroup H H H [37 ] , Theorem 1.11] to H H H G G G
Virtual fibering and ambient embeddings
Agol’s virtual RFRS theorem for right-angled Artin groups, together with the Salvetti embedding and inheritance by subgroups, gives a finite-index RFRS subgroup of G G G [1 ] (Corollary 2.3). In this condition, a descending sequence H ′ = H 0 ′ ≥ H 1 ′ ≥ ⋯ H' = H'_0 \ge H'_1 \ge\cdots H ′ = H 0 ′ ≥ H 1 ′ ≥ ⋯ H ′ H' H ′ H i + 1 ′ H'_{i+1} H i + 1 ′ H i ′ → H 1 ( H i ′ ; Q ) H'_i \to H_1(H'_i;\mathbb{Q}) H i ′ → H 1 ( H i ′ ; Q ) Z \mathbb{Z} Z [37 ] (Section 1.1).
Corollary 18.1. Let G G G
A one-relator group
G = F ( X ) / ⟨ ⟨ r ⟩ ⟩ , G = F(X)/\langle\langle r\rangle\rangle, G = F ( X ) / ⟨⟨ r ⟩⟩ , where X X X r ∈ F ( X ) r \in F(X) r ∈ F ( X ) G G G G G G BS ( m , n ) \operatorname{BS}(m,n) BS ( m , n ) m , n m,n m , n
A word-hyperbolic ascending mapping torus
G = ⟨ F , t ∣ t − 1 f t = ϕ ( f ) ( f ∈ F ) ⟩ , G = \langle F,t \mid t^{-1}ft = \phi(f)\ (f \in F)\rangle, G = ⟨ F , t ∣ t − 1 f t = ϕ ( f ) ( f ∈ F )⟩ , where F F F ϕ : F → F \phi:F \to F ϕ : F → F
There are a finite-index subgroup H ≤ G H \le G H ≤ G d ≥ 0 d \ge0 d ≥ 0 α \alpha α d d d F d F_d F d
H ↪ F d ⋊ α Z . H \hookrightarrow F_d \rtimes_\alpha\mathbb{Z}. H ↪ F d ⋊ α Z . In particular, G G G H H H
1 ⟶ K ⟶ H ⟶ Q ⟶ 1 , K free , Q ≤ Z . 1 \longrightarrow K \longrightarrow H \longrightarrow Q \longrightarrow1,\qquad K\ \text{free},\quad Q \le\mathbb{Z}. 1 ⟶ K ⟶ H ⟶ Q ⟶ 1 , K free , Q ≤ Z . The rank of K K K Q Q Q
Proof. If G G G H = 1 H = 1 H = 1 d = 0 d = 0 d = 0 Z \mathbb{Z} Z G G G
For the first case, a compact connected planar surface with ∣ X ∣ + 1 |X| + 1 ∣ X ∣ + 1 F ( X ) F(X) F ( X ) X = ∅ X = \varnothing X = ∅ G G G [52 ] first makes it hyperbolic. Theorem 1.1 then gives virtual compact specialness. Kielak–Linton’s Corollary 1.3 [37 ] applies to the quotient of this surface group by the normal closure of r r r G G G
In the second case, Corollary 4.10 already gives a finite-index free-by-Z \mathbb{Z} Z [37 ] .
For the stronger embedding, apply Theorem 1.1 of [37 ] to the hyperbolic virtually compact special group G G G Z \mathbb{Z} Z G G G Z \mathbb{Z} Z Z \mathbb{Z} Z 1 ∈ Z 1 \in\mathbb{Z} 1 ∈ Z F d ⋊ α Z F_d \rtimes_\alpha\mathbb{Z} F d ⋊ α Z
Identify H H H π : F d ⋊ α Z → Z \pi: F_d \rtimes_\alpha\mathbb{Z} \to\mathbb{Z} π : F d ⋊ α Z → Z H H H K = H ∩ F d K = H \cap F_d K = H ∩ F d Q = π ( H ) Q = \pi(H) Q = π ( H ) K K K Q Q Q Z \mathbb{Z} Z F d F_d F d K K K
For an ascending torus, one can use the stronger finite-index subgroup from Corollary 4.10 as its own ambient finite-rank free-by-Z \mathbb{Z} Z [51 ] , Overview, p. 3, Question (Virtual fibering)]. The separate quadratic-Dehn-function variant in that source is not covered by the hyperbolicity hypothesis here. Thus a finite-index subgroup is a mapping torus of a finite-rank free-group automorphism, even when the original ascending map is not surjective. This stronger conclusion is specific to the acyclic ascending setting; the general one-relator conclusion retains the possibly infinite-rank kernel and possibly trivial quotient stated above.
