Introduction
A single defining relation can produce very different geometries. The Baumslag–Solitar groups
BS ( m , n ) = ⟨ a , t ∣ t a m t − 1 = a n ⟩ , m , n ∈ Z ∖ { 0 } , \operatorname{BS}(m,n)=\langle a,t\mid ta^{m}t^{-1}=a^{n}\rangle,\qquad m,n\in\mathbb{Z}\setminus\{0\}, BS ( m , n ) = ⟨ a , t ∣ t a m t − 1 = a n ⟩ , m , n ∈ Z ∖ { 0 } , are the basic obstructions to hyperbolicity among one-relator groups. Gersten asked whether they are the only subgroup obstructions [6 ] . We prove the following statement, resolving Gersten’s conjecture positively. Here F ( X ) F(X) F ( X ) X X X ⟨ X ∣ r ⟩ \langle X\mid r\rangle ⟨ X ∣ r ⟩ r r r
Theorem 1.1. Let X X X r ∈ F ( X ) r\in F(X) r ∈ F ( X ) G = ⟨ X ∣ r ⟩ G=\langle X\mid r\rangle G = ⟨ X ∣ r ⟩ BS ( m , n ) \operatorname{BS}(m,n) BS ( m , n ) m , n m,n m , n G G G
The parameters in this statement may have either sign. In particular, the hypothesis excludes BS ( 1 , 1 ) = Z \operatorname{BS}(1,1)=\mathbb{Z} BS ( 1 , 1 ) = Z BS ( 1 , n ) \operatorname{BS}(1,n) BS ( 1 , n ) n n n [5 ] . Theorem 1.1 includes trivial relators and proper powers; it has no torsion-free hypothesis.
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\leq\mathbb{Z} Q ≤ Z [10 ] . The companion theorem [21 ] provides that hypothesis. Combining it with Theorem 1.1 gives the Baumslag–Solitar-free form in Corollary 9.1, together with a finite-index embedding into a free-by-Z \mathbb{Z} Z
Earlier work and the remaining step
The classical theory already separates the torsion case from the main difficulty. Magnus’s Freiheitssatz identifies free subgroups generated by suitable proper subsets of a one-relator generating set [17 ] . Newman’s spelling theorem for proper-power relators gives a Dehn algorithm and hence hyperbolicity [20 ] ; see also the proof in [18 ] . Thus the new hyperbolicity assertion concerns non-power relators, although Theorem 1.1 includes all relators. The Freiheitssatz makes an inductive approach possible: a Magnus splitting has a smaller one-relator vertex group and free edge groups. The geometric form of the Magnus–Moldavanskii–Masters hierarchy, developed by Linton, realizes these splittings through domains in cyclic covers [12 ] . We use that construction in Section 8, with the domain and complexity arguments given in the form needed here.
Several recent results identify substantial classes where this induction preserves hyperbolicity. Louder and Wilton characterized negative immersions for one-relator presentation complexes by primitivity rank greater than two [15 ] . Here the primitivity rank of a relator r r r F ( X ) F(X) F ( X ) r r r r r r ∞ \infty ∞ [15 ] [16 ] . Linton then proved hyperbolicity and virtual specialness for groups with negative immersions [12 ] . His later characterization says that a finitely generated one-relator group is both hyperbolic and locally quasiconvex precisely when its relator has primitivity rank different from two [11 ] . Local quasiconvexity means that every finitely generated subgroup is quasiconvex. This characterization does not settle hyperbolicity alone for Baumslag–Solitar-free groups whose relator has primitivity rank two.
Linton’s hierarchy criterion makes the remaining obstruction more specific: a one-relator hierarchy is quasiconvex with hyperbolic ambient group if and only if it is Z \mathbb{Z} Z [12 ] . The stability hypothesis controls the persistence of noncyclic intersections along the hierarchy; it is an additional condition, not a consequence supplied by that criterion. A complementary reduction to primitive-extension groups isolates a particular class of two-generator one-relator groups in which to settle the conjecture [13 ] . For a broader historical account, see [14 ] .
The geometric issue is expansion in a graph of groups. The Bestvina–Feighn combination theorem makes annular flaring a sufficient condition for hyperbolicity when the vertex and edge data satisfy the required geometric hypotheses [1 ] . We use the algebraic annular formulation in [8 ] . For mapping tori of free-group automorphisms, Brinkmann proved that the absence of periodic conjugacy classes forces expansion and hyperbolicity [3 ] . Mutanguha extended the hyperbolicity criterion to ascending HNN extensions of finite-rank free groups: they are hyperbolic exactly when they contain no Baumslag–Solitar subgroup [19 ] . His expanding relative immersions, iterated pullbacks, and annular arguments supply important precedents for treating persistent intersections [19 ] . The combination theorem here permits both free edge subgroups to be proper in a hyperbolic vertex group, under the intersection and rank hypotheses below.
The combination argument
Our main technical result is a combination theorem for a restricted class of HNN extensions of torsion-free hyperbolic groups (Theorem 3.1). The Magnus construction reduces the problem to a one-relator vertex group of smaller complexity and finitely generated free edge groups. Once the vertex group is hyperbolic, Linton’s quasiconvexity theorem gives quasiconvex edge groups. Two further intersection properties are needed: conjugates of the edge groups have noncyclic intersection only in one specified double coset, and their overlap subgroups satisfy a strong reduced-rank inequality. Collins’s intersection theorem and Linton’s strong-inertia result supply these properties for the Magnus splittings used here [4 , 12 ] . The combination theorem shows that the absence of Baumslag–Solitar subgroups forces annular flaring under precisely these hypotheses.
We prove flaring by contradiction, using free-group currents: invariant measures on pairs of distinct boundary points [9 ] . Long nonflaring annuli give a nonzero stationary current for which almost every line can be transferred across successive edges of the Bass–Serre tree. The stabilizers of increasingly long tree paths are represented by finite labeled core graphs. The rank inequality bounds the number of edges after degree-two vertices are suppressed; it does not bound their label lengths. Along a subsequence, the edges with bounded labels form one fixed graph, called the small subgraph. Its noncyclic components determine finitely many maximal subgroup carriers for persistent path stabilizers. A return along a carrier induces an atoroidal automorphism, with no nontrivial periodic conjugacy class. Brinkmann’s expansion theorem then forces the set of carrier lines to have zero current mass.
Zero mass does not exclude a carrier line from the closed support of the current. This distinction is the main issue in Section 6. Starting with a putative carrier line in the support, we transport a positive-mass cylinder along expanding carrier returns. If this transport continues to follow the prescribed carrier, recentering and compactness concentrate positive mass on its zero-mass line set. If it fails, the stabilizers of the competing actual tree path and the prescribed path have at most cyclic intersection. A sufficiently long overlap consequently follows a periodic axis of uniformly bounded primitive period. Recentering these overlaps puts a periodic line in the support. We reuse this same failure mechanism to exclude periodic support lines: pulling the overlap back along the carrier return preserves enough whole periods to identify the failure axis with the original periodic trajectory, contradicting the competing path choices. Counting whole periods is essential because inverse automorphisms can enlarge primitive period lengths.
It follows that no support line lies in the small subgraph and that every support line has irrational endpoints. The bounded graph complexity now leaves only finitely many rays admitting two different pasts. Transfer permutes their finitely many endpoint orbits under the free edge group. A return at one of these endpoints supplies infinitely many disjoint translates of a fixed positive-mass set inside a compact set, contradicting local finiteness of the current. This proves the restricted combination theorem; the geometric Magnus induction then proves Theorem 1.1.
Magnus subgraphs
We also record the geometric subgroup conclusion in the scope needed by the induction. A circuit in a graph is a map from a subdivided circle; it is immersed when its edge path has no backtracking, including at the closing point. A nontrivial element is a non-power if it is not u k u^k u k ∣ k ∣ ≥ 2 |k| \ge2 ∣ k ∣ ≥ 2
Theorem 1.2. Let Γ \Gamma Γ finite connected graph, and let λ : S 1 → Γ \lambda\colon S^1 \to\Gamma λ : S 1 → Γ π 1 Γ \pi_1\Gamma π 1 Γ λ \lambda λ Y Y Y π 1 Y \pi_1Y π 1 Y S ⊆ Γ S \subseteq\Gamma S ⊆ Γ λ ( S 1 ) ⊈ S \lambda(S^1) \not\subseteq S λ ( S 1 ) ⊆ S
π 1 S ⟶ π 1 Y \pi_1S \longrightarrow\pi_1Y π 1 S ⟶ π 1 Y is injective and its image is quasiconvex.
A choice of basepoint paths changes these subgroups only by conjugacy. Such an S S S Magnus subgraph . Its injectivity follows from the Freiheitssatz. Once Theorem 1.1 gives ambient hyperbolicity, its quasiconvexity follows from Linton’s theorem that Magnus subgroups of hyperbolic one-relator groups are quasiconvex [13 ] . We prove the graph-to-basis identification explicitly in Lemma 8.1. The same quasiconvexity theorem supplies the quasiconvex edge groups at each inductive step, so the induction needs only to establish hyperbolicity of the smaller vertex group.
Structure of the proof
Sections 2–7 prove the restricted combination theorem. Section 3 states its hypotheses and converts nonflaring annuli into a bounded bilateral sequence of currents. Section 4 removes atoms and obtains a stationary current with well-defined transfers on a conull set. Sections 5 and 6 construct the finite carriers and exclude periodic support lines. Section 7 finishes the measure contradiction. Finally, Section 8 constructs a decreasing Magnus induction and proves Theorems 1.1 and 1.2. Section 9 derives the virtual free-by-cyclic consequence and records the corresponding ascending-HNN consequence of the companion theorem.
Intersections, cyclic commensurators, and rank bounds
We need three preliminary tools. The first controls intersections of quasiconvex subgroups. The second produces an embedded Baumslag–Solitar subgroup from a tree action. The third bounds the total reduced rank of a family of conjugate intersections.
Quasiconvex intersections
All word metrics come from finite generating sets. A group is word-hyperbolic if geodesic triangles in a Cayley graph are uniformly thin: each side lies in a fixed neighborhood of the other two. A subgroup of a hyperbolic group is quasiconvex if every geodesic between its elements lies in a fixed neighborhood of the subgroup. We use the Morse lemma and its usual consequences: quasiconvex subgroups are finitely generated and quasi-isometrically embedded, these two properties characterize quasiconvexity, and infinite cyclic subgroups are quasiconvex. In particular, quasiconvexity is transitive between hyperbolic groups, since quasi-isometric embeddings compose. For these background facts, see [2 ] . For a subset Q Q Q Λ Q \Lambda Q Λ Q
Lemma 2.1 (Quasiconvex intersections). Let K K K L L L H H H K ∩ L K \cap L K ∩ L
Λ ( K ∩ L ) = Λ K ∩ Λ L . \Lambda(K \cap L) = \Lambda K \cap\Lambda L. Λ ( K ∩ L ) = Λ K ∩ Λ L . The same assertions hold for any finite family of quasiconvex subgroups.
Proof. First observe that, for fixed subgroups K K K L L L R ≥ 0 R \ge0 R ≥ 0 D D D
N R ( K ) ∩ N R ( L ) ⊆ N D ( K ∩ L ) . N_R(K) \cap N_R(L) \subseteq N_D(K \cap L). N R ( K ) ∩ N R ( L ) ⊆ N D ( K ∩ L ) . Indeed, if x x x k ∈ K k \in K k ∈ K l ∈ L l \in L l ∈ L R R R x x x d = k − 1 l d = k^{-1}l d = k − 1 l 2 R 2R 2 R d d d k d ∈ K , l d ∈ L k_d \in K, l_d \in L k d ∈ K , l d ∈ L k d − 1 l d = d k_d^{-1}l_d = d k d − 1 l d = d
z = k k d − 1 = l l d − 1 ∈ K ∩ L , d ( x , z ) ≤ R + ∣ k d ∣ . z = kk_d^{-1} = ll_d^{-1} \in K \cap L,\qquad d(x,z) \le R + \lvert k_d\rvert. z = k k d − 1 = l l d − 1 ∈ K ∩ L , d ( x , z ) ≤ R + ∣ k d ∣ . Taking a maximum over the finitely many d d d (2.1) .
A geodesic with endpoints in K ∩ L K \cap L K ∩ L K K K L L L (2.1) proves quasiconvexity of their intersection.
Now let ξ ∈ Λ K ∩ Λ L \xi\in\Lambda K \cap\Lambda L ξ ∈ Λ K ∩ Λ L ξ \xi ξ (2.1) along the ray gives a sequence in K ∩ L K \cap L K ∩ L ξ \xi ξ ξ ∈ Λ ( K ∩ L ) \xi\in\Lambda(K \cap L) ξ ∈ Λ ( K ∩ L ) [7 ] , Lemmas 2.6 and 2.7.
Lemma 2.2 (Finite double-coset lists). Let K K K L L L H H H K g L KgL K g L K ∩ g L g − 1 K \cap gLg^{-1} K ∩ g L g − 1 H H H
Proof. Suppose that c ∈ K ∩ g L g − 1 c \in K \cap gLg^{-1} c ∈ K ∩ g L g − 1 Λ K \Lambda K Λ K Λ ( g L ) \Lambda(gL) Λ ( g L ) c n ∈ K c^n \in K c n ∈ K c n g ∈ g L c^n g \in gL c n g ∈ g L n n n K K K g L gL g L q q q q q q g L gL g L L L L
At any point of this line choose nearby vertices k ∈ K k \in K k ∈ K g l ∈ g L gl \in gL g l ∈ g L k − 1 g l ∈ K g L k^{-1}gl \in KgL k − 1 g l ∈ K g L
The qualification about infinite-order elements is necessary in groups with torsion. For example, the central subgroup C 2 C_2 C 2 F 2 × C 2 F_2 \times C_2 F 2 × C 2 C 2 C_2 C 2
For later applications to cosets, note also that
Λ ( g L ) = Λ ( g L g − 1 ) . \Lambda(gL) = \Lambda(gLg^{-1}). Λ ( g L ) = Λ ( g L g − 1 ) . Indeed, right multiplication by the fixed element g − 1 g^{-1} g − 1 g L gL g L
Commensuration in a tree action
Two subgroups are commensurable if their intersection has finite index in both. For a subgroup C ≤ H C \le H C ≤ H
Comm H ( C ) = { h ∈ H : C and h C h − 1 are commensurable } . \operatorname{Comm}_{H}(C) = \{h \in H : C \text{ and } hCh^{-1} \text{ are commensurable}\}. Comm H ( C ) = { h ∈ H : C and h C h − 1 are commensurable } . This is a subgroup. We first deduce the cyclic case directly from the double-coset lemma.
Lemma 2.3. If C C C H H H C C C Comm H ( C ) \operatorname{Comm}_{H}(C) Comm H ( C ) H H H
Proof. The subgroup C C C C g C CgC C g C g g g C g C CgC C g C [ C : C ∩ g − 1 C g ] [C:C \cap g^{-1}Cg] [ C : C ∩ g − 1 C g ] C C C
All tree actions below are actions on simplicial trees without inversions. An element is elliptic if it fixes a vertex and tree-hyperbolic if it translates a bi-infinite geodesic. We will need an embedded Baumslag–Solitar group, rather than just a power-conjugacy relation. The following normal-form argument provides it.
Lemma 2.4 (Graphs of cyclic groups). Let Q Q Q Q Q Q B S ( m , n ) BS(m,n) B S ( m , n ) m , n m,n m , n
Proof. Choose a vertex v v v S S S Q Q Q Q Q Q
D = ⋃ s ∈ S [ v , s v ] D = \bigcup_{s \in S} [v,sv] D = s ∈ S ⋃ [ v , s v ] is connected: its segments connect the vertices in the orbit Q v Qv Q v Q Q Q Q v Qv Q v D D D Q Q Q
Collapse any non-loop edge whose map to one endpoint group is onto. Such a collapse preserves the fundamental group and leaves a graph of infinite cyclic groups. After finitely many collapses, every remaining non-loop edge has both inclusion indices greater than one.
If a loop remains, the subgraph consisting of that loop and its vertex has fundamental group
⟨ x , t ∣ t x m t − 1 = x n ⟩ = B S ( m , n ) , m , n ≠ 0. \langle x,t \mid tx^{m}t^{-1}=x^{n}\rangle= BS(m,n), \qquad m,n \ne0. ⟨ x , t ∣ t x m t − 1 = x n ⟩ = B S ( m , n ) , m , n = 0. Its fundamental group injects into Q Q Q
If a non-loop edge remains, its two-vertex subgraph likewise injects and has fundamental group
⟨ x , y ∣ x m = y n ⟩ , ∣ m ∣ , ∣ n ∣ > 1. \langle x,y \mid x^{m}=y^{n}\rangle, \qquad|m|,|n|>1. ⟨ x , y ∣ x m = y n ⟩ , ∣ m ∣ , ∣ n ∣ > 1. The element z = x m = y n z=x^{m}=y^{n} z = x m = y n ⟨ z ⟩ \langle z\rangle ⟨ z ⟩ C ∣ m ∣ ∗ C ∣ n ∣ C_{|m|} * C_{|n|} C ∣ m ∣ ∗ C ∣ n ∣ x y xy x y z z z x y xy x y Z 2 = B S ( 1 , 1 ) \mathbb{Z}^{2}=BS(1,1) Z 2 = B S ( 1 , 1 )
Finally, if no edge remains, the graph has one vertex and Q Q Q
Lemma 2.5 (Elliptic commensuration). Suppose that G G G T T T a ∈ G a \in G a ∈ G g ∈ G g \in G g ∈ G G G G B S ( m , n ) BS(m,n) B S ( m , n ) m , n m,n m , n
Proof. Set L = ⟨ a , g ⟩ L=\langle a,g\rangle L = ⟨ a , g ⟩ v v v a a a L L L ⟨ a ⟩ \langle a\rangle ⟨ a ⟩ h ∈ L h \in L h ∈ L r , s r,s r , s a r = h a s h − 1 a^{r}=ha^{s}h^{-1} a r = h a s h − 1 a a a h v hv h v L v Lv Lv T L = Hull ( L v ) T_L=\operatorname{Hull}(Lv) T L = Hull ( Lv ) a a a
Let x x x T L T_L T L N > 0 N>0 N > 0 a N ∈ L x a^{N} \in L_x a N ∈ L x L x L_x L x ⟨ a N ⟩ \langle a^{N}\rangle ⟨ a N ⟩ G x G_x G x
⟨ a N ⟩ ≤ L x ≤ Comm G x ( ⟨ a N ⟩ ) . \langle a^{N}\rangle\leq L_x \leq\operatorname{Comm}_{G_x}(\langle a^{N}\rangle). ⟨ a N ⟩ ≤ L x ≤ Comm G x (⟨ a N ⟩) . Lemma 2.3 makes the last group infinite cyclic, so L x L_x L x T L T_L T L a a a L L L
The group L L L a = c r a=c^r a = c r g = c s g=c^s g = c s ℓ ( a ) = ∣ r ∣ ℓ ( c ) = 0 \ell(a)=|r|\ell(c)=0 ℓ ( a ) = ∣ r ∣ ℓ ( c ) = 0 ℓ ( g ) = 0 \ell(g)=0 ℓ ( g ) = 0 g g g L L L T L T_L T L
Strong inertia in free groups
We now formulate the rank estimate that will control noncyclic intersections. For a finitely generated free group K K K
r ˉ ( K ) = max { rk ( K ) − 1 , 0 } . \bar{r}(K)=\max\{\operatorname{rk}(K)-1,0\}. r ˉ ( K ) = max { rk ( K ) − 1 , 0 } . Strong inertia bounds the sum of reduced ranks of all conjugate intersections; see [12 ] .
Definition 2.6. A finitely generated subgroup E E E P P P strongly inert if, for every finitely generated subgroup S ≤ P S\leq P S ≤ P
∑ S p E ∈ S \ P / E r ˉ ( E ∩ p − 1 S p ) ≤ r ˉ ( S ) . (1) \sum_{SpE\in S\backslash P/E}\bar{r}(E\cap p^{-1}Sp)\leq\bar{r}(S). \tag*{(1)} S pE ∈ S \ P / E ∑ r ˉ ( E ∩ p − 1 S p ) ≤ r ˉ ( S ) . (1) The summands are independent of the chosen double-coset representatives. Only finitely many intersections are nontrivial, by Lemma 2.2.
Corollary 2.7. Let P P P E ≤ P E\leq P E ≤ P strongly inert subgroup. If g ∈ P ∖ E g\in P\setminus E g ∈ P ∖ E E ∩ g E g − 1 E\cap gEg^{-1} E ∩ g E g − 1
Proof. Apply (2.2) with S = E S=E S = E r ˉ ( E ) \bar{r}(E) r ˉ ( E )
Annuli and currents for a restricted HNN extension
We now isolate the combination statement used in the Magnus induction. The restrictions on intersections in its hypotheses will provide a uniform bound on the ranks of path stabilizers. In this section we establish that failure of hyperbolicity produces a bounded, nonzero trajectory of free-group currents. The subsequent sections rule out that trajectory and thereby complete the combination argument.
Theorem 3.1 (Restricted combination theorem). Let H H H A , B ≤ H A,B\leq H A , B ≤ H ϕ : A → B \phi:A\to B ϕ : A → B
G = ⟨ H , t ∣ t a t − 1 = ϕ ( a ) ( a ∈ A ) ⟩ , P = A , I = A ∩ B , J = t − 1 I t ≤ P . G=\langle H,t\mid tat^{-1}=\phi(a)\ (a\in A)\rangle,\qquad P=A,\qquad I=A\cap B,\qquad J=t^{-1}It\leq P. G = ⟨ H , t ∣ t a t − 1 = ϕ ( a ) ( a ∈ A )⟩ , P = A , I = A ∩ B , J = t − 1 I t ≤ P . Assume the following two conditions.
(i) For M , N ∈ { A , B } M,N\in\{A,B\} M , N ∈ { A , B } g ∈ H g\in H g ∈ H M ∩ g N g − 1 M\cap gNg^{-1} M ∩ g N g − 1 g ∈ M N g\in MN g ∈ M N
(ii) The groups I , J I,J I , J E ∈ { I , J } E\in\{I,J\} E ∈ { I , J }
∑ S p E ∈ S \ P / E r ˉ ( E ∩ p − 1 S p ) ≤ r ˉ ( S ) for every finitely generated S ≤ P . (2) \sum_{SpE\in S\backslash P/E}\bar{r}(E\cap p^{-1}Sp)\leq\bar{r}(S)\qquad\text{for every finitely generated }S\leq P. \tag*{(2)} S pE ∈ S \ P / E ∑ r ˉ ( E ∩ p − 1 S p ) ≤ r ˉ ( S ) for every finitely generated S ≤ P . (2) If G G G B S ( m , n ) BS(m,n) B S ( m , n ) m , n m,n m , n G G G
All the intersection restrictions in this statement are hypotheses. In particular, the assertion concerns a restricted class of HNN extensions. We retain the notation of Theorem 3.1 throughout the combination argument. Write
i : I ⟶ P , i ( c ) = c , j : I ⟶ P , j ( c ) = t − 1 c t . i:I\longrightarrow P,\quad i(c)=c,\qquad j:I\longrightarrow P,\quad j(c)=t^{-1}ct. i : I ⟶ P , i ( c ) = c , j : I ⟶ P , j ( c ) = t − 1 c t . so that j ( I ) = J j(I)=J j ( I ) = J H H H P P P ∣ ⋅ ∣ H |\cdot|_{H} ∣ ⋅ ∣ H ∣ ⋅ ∣ P |\cdot|_{P} ∣ ⋅ ∣ P P = 1 P=1 P = 1 G = H ∗ Z G=H*\mathbb{Z} G = H ∗ Z P ≠ 1 P\ne1 P = 1
Frames and stabilizers of paths
Let T \mathcal{T} T G → Z G\to\mathbb{Z} G → Z t t t 1 1 1 H H H 0 0 0 e e e t − 1 H t^{-1}H t − 1 H H H H P P P frame for the edge h e he h e h ∈ G h\in G h ∈ G h h h h p hp h p p ∈ P p\in P p ∈ P
Lemma 3.2 (Neighboring edges and frames). A reduced path in T \mathcal{T} T e e e p t e pte pt e p I ∈ P / I pI\in P/I p I ∈ P / I p t − 1 e pt^{-1}e p t − 1 e p J ∈ P / J pJ\in P/J p J ∈ P / J e e e
p I p − 1 and p J p − 1 . pIp^{-1}\qquad\text{and}\qquad pJp^{-1}. p I p − 1 and p J p − 1 . For a positive successor the coordinate transfer is c ↦ t − 1 p − 1 c p t c\mapsto t^{-1}p^{-1}cpt c ↦ t − 1 p − 1 c pt c ↦ t p − 1 c p t − 1 c\mapsto tp^{-1}cpt^{-1} c ↦ t p − 1 c p t − 1
Proof. Every positively oriented edge leaving H H H a t e ate a t e a ∈ H a\in H a ∈ H a B ∈ H / B aB\in H/B a B ∈ H / B e e e A ∩ a B a − 1 A\cap aBa^{-1} A ∩ a B a − 1 a ∈ A B a\in AB a ∈ A B a = p b a=pb a = p b p ∈ P p\in P p ∈ P b ∈ B b\in B b ∈ B b t = t φ − 1 ( b ) bt=t\varphi^{-1}(b) b t = t φ − 1 ( b ) a t e ate a t e p t e pte pt e
p t e = q t e ⟺ t − 1 q − 1 p t ∈ A ⟺ q − 1 p ∈ A ∩ B = I . pte=qte\quad\Longleftrightarrow\quad t^{-1}q^{-1}pt\in A\quad\Longleftrightarrow\quad q^{-1}p\in A\cap B=I. pt e = q t e ⟺ t − 1 q − 1 pt ∈ A ⟺ q − 1 p ∈ A ∩ B = I . The common stabilizer is P ∩ p B p − 1 = p I p − 1 P\cap pBp^{-1}=pIp^{-1} P ∩ pB p − 1 = p I p − 1 ( p t ) − 1 (pt)^{-1} ( pt ) − 1
A positive predecessor ending at t − 1 H t^{-1}H t − 1 H t − 1 a e t^{-1}ae t − 1 a e a A ∈ H / A aA\in H/A a A ∈ H / A t t t B ∩ a A a − 1 B\cap aAa^{-1} B ∩ a A a − 1 a ∈ B A a\in BA a ∈ B A a = b r a=br a = b r p = t − 1 b t ∈ P p=t^{-1}bt\in P p = t − 1 b t ∈ P p t − 1 e pt^{-1}e p t − 1 e
q − 1 p ∈ P ∩ t − 1 P t = t − 1 ( B ∩ A ) t = J . q^{-1}p\in P\cap t^{-1}Pt=t^{-1}(B\cap A)t=J. q − 1 p ∈ P ∩ t − 1 P t = t − 1 ( B ∩ A ) t = J . Their common stabilizer with e e e p J p − 1 pJp^{-1} p J p − 1 ( p t − 1 ) − 1 (pt^{-1})^{-1} ( p t − 1 ) − 1
At a reversal of traversal sign, the consecutive edge stabilizers, in the common vertex coordinates, are M M M a M a − 1 aMa^{-1} a M a − 1 M ∈ { A , B } M\in\{A,B\} M ∈ { A , B } a ∉ M a\notin M a ∈ / M
We call a consecutive positive pair aligned when its bridge lies in the double coset A B AB A B
Lemma 3.3 (Rank bound for a fixed window). Fix integers ℓ , r ≥ 0 \ell, r \ge0 ℓ , r ≥ 0 − ℓ , … , r -\ell,\ldots,r − ℓ , … , r e e e
∑ [ W ] ∈ P \ { such paths } r ˉ ( Stab G ( W ) ) ≤ r ˉ ( P ) . (3) \sum_{[W]\in P\backslash\{\text{such paths}\}} \bar r(\operatorname{Stab}_{G}(W)) \le\bar r(P). \tag*{(3)} [ W ] ∈ P \ { such paths } ∑ r ˉ ( Stab G ( W )) ≤ r ˉ ( P ) . (3) Every stabilizer in this sum is finitely generated and free. In particular, there are at most r ˉ ( P ) \bar r(P) r ˉ ( P ) P P P G G G
Proof. Build the fixed window by starting with e e e P P P W W W S = Stab G ( W ) ≤ P S = \operatorname{Stab}_{G}(W) \le P S = Stab G ( W ) ≤ P h h h K = h − 1 S h ≤ P K = h^{-1}Sh \le P K = h − 1 S h ≤ P
For a forward extension whose stabilizer is noncyclic, Lemma 3.2 restricts the next edge to h p I e hpIe h p I e p I ∈ P / I pI \in P/I p I ∈ P / I S S S K K K W W W K p I KpI K p I K ∩ p I p − 1 K \cap pIp^{-1} K ∩ p I p − 1
∑ K p I r ˉ ( K ∩ p I p − 1 ) ≤ r ˉ ( K ) = r ˉ ( S ) . \sum_{KpI} \bar r(K \cap pIp^{-1}) \le\bar r(K) = \bar r(S). K p I ∑ r ˉ ( K ∩ p I p − 1 ) ≤ r ˉ ( K ) = r ˉ ( S ) . For backward extension use J J J P P P
The forgetful map from extended windows to the preceding windows is P P P W W W S S S Stab G ( e ) = P \operatorname{Stab}_{G}(e) = P Stab G ( e ) = P
The shape ( ℓ , r ) (\ell, r) ( ℓ , r )
Lemma 3.4. Two distinct aligned paths of the same fixed window shape with distinguished edge e e e
Proof. At their first divergence from the common part, the common stabilizer, in the shared endpoint frame, lies in
p E p − 1 ∩ q E q − 1 , p E ≠ q E , E ∈ { I , J } . pEp^{-1} \cap qEq^{-1}, \qquad pE \ne qE, \qquad E \in\{I,J\}. pE p − 1 ∩ q E q − 1 , pE = q E , E ∈ { I , J } . This intersection is at most cyclic. Indeed, apply (2) with S = E S = E S = E r ˉ ( E ) \bar r(E) r ˉ ( E ) p − 1 q ∉ E p^{-1}q \notin E p − 1 q ∈ / E E E E
The algebraic annular-flaring criterion
We state the external combination theorem with its quantifiers before passing to limits. This is important because the width bound in a sequence of counterexamples will depend on the number of rows.
For a traversal sign ϵ ∈ { + , − } \epsilon\in\{+,-\} ϵ ∈ { + , − } H H H
f + ( c ) = c , f − ( c ) = t c t − 1 , l + ( c ) = t c t − 1 , l − ( c ) = c . f_{+}(c)=c,\qquad f_{-}(c)=tct^{-1},\qquad l_{+}(c)=tct^{-1},\qquad l_{-}(c)=c. f + ( c ) = c , f − ( c ) = t c t − 1 , l + ( c ) = t c t − 1 , l − ( c ) = c . An algebraic annulus with 2 m + 1 2m+1 2 m + 1 ϵ s ∈ { + , − } \epsilon_s \in\{+,-\} ϵ s ∈ { + , − } c s ∈ P ∖ { 1 } c_s \in P \setminus\{1\} c s ∈ P ∖ { 1 } − m ≤ s ≤ m -m \le s \le m − m ≤ s ≤ m a s ∈ H a_s \in H a s ∈ H − m ≤ s < m -m \le s < m − m ≤ s < m
f ϵ s ( c s ) = a s l ϵ s + 1 ( c s + 1 ) a s − 1 . f_{\epsilon_s}(c_s)=a_s l_{\epsilon_{s+1}}(c_{s+1})a_s^{-1}. f ϵ s ( c s ) = a s l ϵ s + 1 ( c s + 1 ) a s − 1 . Its width is max s ∣ a s ∣ H \max_s |a_s|_H max s ∣ a s ∣ H ∣ c 0 ∣ P |c_0|_P ∣ c 0 ∣ P essential if a + , − +,- + , − a s ∉ A a_s \notin A a s ∈ / A − , + -,+ − , + a s ∉ B a_s \notin B a s ∈ / B λ \lambda λ hyperbolic if
max { ∣ c − m ∣ P , ∣ c m ∣ P } ≥ λ ∣ c 0 ∣ P . \max\{|c_{-m}|_P,|c_m|_P\}\geq\lambda|c_0|_P. max { ∣ c − m ∣ P , ∣ c m ∣ P } ≥ λ ∣ c 0 ∣ P . All these lengths are based word lengths.
For completeness, these definitions describe an actual reduced tree path. Write f ϵ s ( c ) = u − 1 c u f_{\epsilon_s}(c)=u^{-1}cu f ϵ s ( c ) = u − 1 c u l ϵ s + 1 ( c ) = v − 1 c v l_{\epsilon_{s+1}}(c)=v^{-1}cv l ϵ s + 1 ( c ) = v − 1 c v u , v ∈ { 1 , t − 1 } u,v\in\{1,t^{-1}\} u , v ∈ { 1 , t − 1 } h s h_s h s
h s + 1 = h s u a s v − 1 . h_{s+1}=h_s u a_s v^{-1}. h s + 1 = h s u a s v − 1 . The two edges meet at h s u H = h s + 1 v H h_suH=h_{s+1}vH h s u H = h s + 1 v H h s c s h s − 1 = h s + 1 c s + 1 h s + 1 − 1 h_sc_sh_s^{-1}=h_{s+1}c_{s+1}h_{s+1}^{-1} h s c s h s − 1 = h s + 1 c s + 1 h s + 1 − 1
( ϵ s , ϵ s + 1 ) ( + , + ) ( + , − ) ( − , + ) ( − , − ) u a s v − 1 a s t a s t − 1 a s t t − 1 a s \begin{array}{c|cccc}
(\epsilon_s,\epsilon_{s+1}) & (+,+) & (+,-) & (-,+) & (-,-)\\
u a_s v^{-1} & a_st & a_s & t^{-1}a_st & t^{-1}a_s
\end{array} ( ϵ s , ϵ s + 1 ) u a s v − 1 ( + , + ) a s t ( + , − ) a s ( − , + ) t − 1 a s t ( − , − ) t − 1 a s At the two reversing transitions the consecutive geometric edges agree exactly when a s ∈ A a_s\in A a s ∈ A a s ∈ B a_s\in B a s ∈ B
Theorem 3.5 (Algebraic annular flaring). Let H H H A , B ≤ H A,B\leq H A , B ≤ H ϕ : A → B \phi:A\to B ϕ : A → B A → H A\to H A → H G = ⟨ H , t ∣ t a t − 1 = ϕ ( a ) , a ∈ A ⟩ G=\langle H,t\mid tat^{-1}=\phi(a),\ a\in A\rangle G = ⟨ H , t ∣ t a t − 1 = ϕ ( a ) , a ∈ A ⟩ G G G λ > 1 \lambda>1 λ > 1 m ≥ 1 m\geq1 m ≥ 1 ρ > 0 \rho>0 ρ > 0 h ( ρ ) h(\rho) h ( ρ ) 2 m + 1 2m+1 2 m + 1 ρ \rho ρ h ( ρ ) h(\rho) h ( ρ ) λ \lambda λ
This is the algebraic formulation of the Bestvina–Feighn combination theorem in [1 ] ; we use its explicit annular statement in [8 ] . The latter numbers refer to the arXiv version of that paper. Our 2 m + 1 2m+1 2 m + 1 A A A B B B H H H
Lemma 3.6 (Counterannuli at a fixed length). If G G G m ≥ 1 m\geq1 m ≥ 1 2 m + 1 2m+1 2 m + 1
∣ c 0 ∣ P ⟶ ∞ , ∣ c − m ∣ P , ∣ c m ∣ P < 2 ∣ c 0 ∣ P . |c_0|_P\longrightarrow\infty,\qquad|c_{-m}|_P,|c_m|_P<2|c_0|_P. ∣ c 0 ∣ P ⟶ ∞ , ∣ c − m ∣ P , ∣ c m ∣ P < 2∣ c 0 ∣ P . For each fixed s ∈ [ − m , m ] s\in[-m,m] s ∈ [ − m , m ] ∣ c s ∣ P / ∣ c 0 ∣ P |c_s|_P/|c_0|_P ∣ c s ∣ P /∣ c 0 ∣ P m m m
Proof. Negate Theorem 3.5 with λ = 2 \lambda= 2 λ = 2 m m m ρ m \rho_m ρ m ρ m \rho_m ρ m
The maps f ϵ , l ϵ f_\epsilon, l_\epsilon f ϵ , l ϵ L ≥ 1 L \ge1 L ≥ 1 D ≥ 0 D \ge0 D ≥ 0
∣ c s + 1 ∣ P ≤ L ∣ c s ∣ P + D , ∣ c s ∣ P ≤ L ∣ c s + 1 ∣ P + D |c_{s+1}|_P \le L|c_s|_P + D,\qquad|c_s|_P \le L|c_{s+1}|_P + D ∣ c s + 1 ∣ P ≤ L ∣ c s ∣ P + D , ∣ c s ∣ P ≤ L ∣ c s + 1 ∣ P + D at every row of this fixed sequence. Iterating from row 0 bounds all the indicated ratios, since ∣ c 0 ∣ P → ∞ |c_0|_P \to\infty ∣ c 0 ∣ P → ∞
Currents and induction along an injection
Let F F F T F T_F T F
∂ 2 F = ( ∂ F × ∂ F ) ∖ { ( ξ , ξ ) : ξ ∈ ∂ F } \partial^2 F = (\partial F \times\partial F) \setminus\{(\xi,\xi):\xi\in\partial F\} ∂ 2 F = ( ∂ F × ∂ F ) ∖ {( ξ , ξ ) : ξ ∈ ∂ F } parametrizes oriented geodesic lines: ( ξ , ζ ) (\xi,\zeta) ( ξ , ζ ) ξ \xi ξ ζ \zeta ζ γ \gamma γ T F T_F T F Cyl F ( γ ) \operatorname{Cyl}_F(\gamma) Cyl F ( γ ) L F ( v ) \mathcal{L}_F(v) L F ( v ) v v v v v v
A current on F F F F F F ∂ 2 F \partial^2 F ∂ 2 F Curr ( F ) \operatorname{Curr}(F) Curr ( F )
∥ μ ∥ F = 1 2 μ ( L F ( 1 ) ) . (4) \|\mu\|_F = \frac{1}{2}\mu(\mathcal{L}_F(1)). \tag*{(4)} ∥ μ ∥ F = 2 1 μ ( L F ( 1 )) . (4) This norm is additive on the cone. It vanishes only on the zero current, since translates of L F ( 1 ) \mathcal{L}_F(1) L F ( 1 ) ∂ 2 F \partial^2 F ∂ 2 F
For w ∈ F ∖ { 1 } w \in F \setminus\{1\} w ∈ F ∖ { 1 } counting current η w \eta_w η w w w w F / ⟨ w ⟩ F/\langle w\rangle F / ⟨ w ⟩ ⟨ w ⟩ \langle w\rangle ⟨ w ⟩ w w w 1 1 1
∥ η w ∥ F = ∣ w ∣ F , c y c , \|\eta_w\|_F = |w|_{F,\mathrm{cyc}}, ∥ η w ∥ F = ∣ w ∣ F , cyc , where the right side is cyclic word length. More generally, the coordinate on a segment labeled by a reduced word u u u u u u u − 1 u^{-1} u − 1 w w w z z z ∣ z ∣ |z| ∣ z ∣ ∣ u ∣ > ∣ z ∣ |u| > |z| ∣ u ∣ > ∣ z ∣
Lemma 3.7 (Compactness). The norm is continuous on Curr ( F ) \operatorname{Curr}(F) Curr ( F )
Proof. Continuity follows because L F ( 1 ) \mathcal{L}_F(1) L F ( 1 ) ∂ 2 F \partial^2 F ∂ 2 F L F ( v ) \mathcal{L}_F(v) L F ( v )
We use the current map associated to an injection with its full induction over cosets. An ordinary boundary push alone would generally fail to be invariant under the target group. The construction below agrees with [9 ] (Proposition–Definition 12.1) in the nonabelian case, and also covers rank one.
Lemma 3.8 (Induction of currents). Let θ : Q → F \theta: Q \to F θ : Q → F nontrivial finite-rank free groups . It defines a continuous linear map
θ ∗ : Curr ( Q ) ⟶ Curr ( F ) \theta_*: \operatorname{Curr}(Q) \longrightarrow\operatorname{Curr}(F) θ ∗ : Curr ( Q ) ⟶ Curr ( F ) which sends η w \eta_w η w η θ ( w ) \eta_{\theta(w)} η θ ( w ) 0 < c θ ≤ C θ < ∞ 0 < c_\theta\le C_\theta< \infty 0 < c θ ≤ C θ < ∞
c θ ∥ ν ∥ Q ≤ ∥ θ ∗ ν ∥ F ≤ C θ ∥ ν ∥ Q ( ν ∈ Curr ( Q ) ) . (5) c_\theta\lVert\nu\rVert_Q \le\lVert\theta_*\nu\rVert_F \le C_\theta\lVert\nu\rVert_Q \qquad(\nu\in\operatorname{Curr}(Q)). \tag*{(5)} c θ ∥ ν ∥ Q ≤ ∥ θ ∗ ν ∥ F ≤ C θ ∥ ν ∥ Q ( ν ∈ Curr ( Q )) . (5) Postcomposing θ \theta θ F F F θ ∗ \theta_* θ ∗ ψ : F → F ′ \psi:F \to F' ψ : F → F ′
( ψ ∘ θ ) ∗ = ψ ∗ ∘ θ ∗ . (\psi\circ\theta)_* = \psi_* \circ\theta_*. ( ψ ∘ θ ) ∗ = ψ ∗ ∘ θ ∗ . Proof. Put E = θ ( Q ) E = \theta(Q) E = θ ( Q ) E E E F F F θ \theta θ ∂ Q → Λ E ⊂ ∂ F \partial Q \to\Lambda E \subset\partial F ∂ Q → Λ E ⊂ ∂ F ν \nu ν E E E ν ^ \widehat{\nu} ν ∂ 2 Λ E \partial^2\Lambda E ∂ 2 Λ E ∂ 2 F \partial^2 F ∂ 2 F
θ ∗ ν = ∑ p E ∈ F / E p ∗ ν ^ . (6) \theta_*\nu= \sum_{pE\in F/E} p_*\widehat{\nu}. \tag*{(6)} θ ∗ ν = pE ∈ F / E ∑ p ∗ ν . (6) The summand depends only on the coset. It remains to prove that this sum is locally finite and continuous.
Let T E T_E T E Λ E \Lambda E Λ E T F T_F T F E E E E E E v v v p E pE pE v ∈ p T E v \in pT_E v ∈ p T E E E E p − 1 v p^{-1}v p − 1 v T E T_E T E (6) that meets a segment cylinder has its hull containing that segment. Hence only finitely many terms contribute to a fixed cylinder. The preimage of that cylinder in ∂ 2 Q \partial^2 Q ∂ 2 Q ∂ 2 Λ E \partial^2\Lambda E ∂ 2 Λ E ∂ 2 F \partial^2 F ∂ 2 F ν \nu ν F F F
The coset decompositions in (6) combine the index sets F / E F/E F / E E / θ ( ⟨ w ⟩ ) E/\theta(\langle w\rangle) E / θ (⟨ w ⟩) F / ⟨ θ ( w ) ⟩ F/\langle\theta(w)\rangle F / ⟨ θ ( w )⟩ ψ ∗ θ ∗ ν \psi_*\theta_*\nu ψ ∗ θ ∗ ν F ′ / ψ ( F ) F'/\psi(F) F ′ / ψ ( F ) ψ ( F ) / ψ θ ( Q ) \psi(F)/\psi\theta(Q) ψ ( F ) / ψ θ ( Q ) F ′ / ψ θ ( Q ) F'/\psi\theta(Q) F ′ / ψ θ ( Q ) ( ψ ∘ θ ) ∗ ν (\psi\circ\theta)_*\nu ( ψ ∘ θ ) ∗ ν
A nonzero measure has nonzero boundary push, and the summands are nonnegative, so its induction is nonzero. By Lemma 3.7, the norm-one set in Curr ( Q ) \operatorname{Curr}(Q) Curr ( Q ) ν ↦ ∥ θ ∗ ν ∥ F \nu\mapsto\lVert\theta_*\nu\rVert_F ν ↦ ∥ θ ∗ ν ∥ F (5) . Finally, conjugating E E E F F F
No injectivity of the linear map θ ∗ \theta_* θ ∗
From based words to currents
The girth of an annulus uses based length, whereas the norm of a counting current uses cyclic length. A conjugating tail can make these lengths very different. We bridge that difference by capping the word in the intersection subgroup before applying either of its two embeddings.
For reduced words u u u w w w N u ( w ) N_u(w) N u ( w ) u u u w w w N u s y m ( w ) = N u ( w ) + N u − 1 ( w ) N_u^{\mathrm{sym}}(w) = N_u(w) + N_{u^{-1}}(w) N u sym ( w ) = N u ( w ) + N u − 1 ( w )
Lemma 3.9 (Capping linear words). Fix an injection θ : Q → F \theta: Q \to F θ : Q → F W W W Q Q Q b b b W b Wb W b u u u F F F D θ , u D_{\theta,u} D θ , u W W W
∣ η θ ( W b ) ( Cyl F ( [ 1 , u ] ) ) − N u s y m ( red ( θ ( W ) ) ) ∣ ≤ D θ , u . (7) \left|\eta_{\theta(Wb)}\bigl(\operatorname{Cyl}_F([1,u])\bigr)-N_u^{\mathrm{sym}}\bigl(\operatorname{red}(\theta(W))\bigr)\right| \le D_{\theta,u}. \tag*{(7)} η θ ( W b ) ( Cyl F ([ 1 , u ]) ) − N u sym ( red ( θ ( W )) ) ≤ D θ , u . (7) There is also a constant D θ D_\theta D θ W W W
∣ ∥ η θ ( W b ) ∥ F − ∣ θ ( W ) ∣ F ∣ ≤ D θ . (8) \left|\|\eta_{\theta(Wb)}\|_F-|\theta(W)|_F\right| \le D_\theta. \tag*{(8)} ∥ η θ ( W b ) ∥ F − ∣ θ ( W ) ∣ F ≤ D θ . (8) Here red \operatorname{red} red b b b Q Q Q
Proof. If Q Q Q b b b b b b W W W W b Wb W b Q Q Q W W W θ \theta θ
The image in T F T_F T F T Q T_Q T Q θ \theta θ W b Wb W b T Q T_Q T Q 1 1 1 1 1 1 θ ( W b ) \theta(Wb) θ ( W b ) D D D W W W 1 1 1 θ ( W b ) \theta(Wb) θ ( W b ) T F T_F T F
red ( θ ( W b ) ) = v z v − 1 \operatorname{red}(\theta(Wb))=v z v^{-1} red ( θ ( W b )) = v z v − 1 with z z z ∣ v ∣ F ≤ D |v|_F \le D ∣ v ∣ F ≤ D
Multiplication by θ ( b ) \theta(b) θ ( b ) red ( θ ( W ) ) \operatorname{red}(\theta(W)) red ( θ ( W )) max { ∣ θ ( x ) ∣ F : x a basis letter of Q } \max\{|\theta(x)|_F : x\text{ a basis letter of }Q\} max { ∣ θ ( x ) ∣ F : x a basis letter of Q } v v v v − 1 v^{-1} v − 1 2 D 2D 2 D u u u z z z ∣ u ∣ − 1 |u|-1 ∣ u ∣ − 1 ∣ z ∣ < ∣ u ∣ |z|<|u| ∣ z ∣ < ∣ u ∣ ∣ z ∣ F = ∣ θ ( W ) ∣ F + O θ ( 1 ) |z|_F=|\theta(W)|_F+O_\theta(1) ∣ z ∣ F = ∣ θ ( W ) ∣ F + O θ ( 1 )
Lemma 3.10 (A current limit for each finite window). Suppose G G G m ≥ 1 m \ge1 m ≥ 1 μ s ∈ Curr ( P ) \mu_s \in\operatorname{Curr}(P) μ s ∈ Curr ( P ) − m ≤ s ≤ m -m \le s \le m − m ≤ s ≤ m
∥ μ 0 ∥ = 1 , ∥ μ − m ∥ , ∥ μ m ∥ ≤ 2 , (9) \lVert\mu_0 \rVert= 1,\qquad\lVert\mu_{-m} \rVert,\lVert\mu_m \rVert\le2, \tag*{(9)} ∥ μ 0 ∥ = 1 , ∥ μ − m ∥ , ∥ μ m ∥ ≤ 2 , (9) and a fixed essential sign and bridge tuple such that each consecutive pair is induced by a current on its intersection subgroup. More precisely, for
K s = f ε s ( P ) ∩ a s l ε s + 1 ( P ) a s − 1 K_s=f_{\varepsilon_s}(P)\cap a_s l_{\varepsilon_{s+1}}(P)a_s^{-1} K s = f ε s ( P ) ∩ a s l ε s + 1 ( P ) a s − 1 there is ν s ∈ Curr ( K s ) \nu_s \in\operatorname{Curr}(K_s) ν s ∈ Curr ( K s )
( θ s , 0 ) ∗ ν s = μ s , ( θ s , 1 ) ∗ ν s = μ s + 1 , (\theta_{s,0})_*\nu_s=\mu_s,\qquad(\theta_{s,1})_*\nu_s=\mu_{s+1}, ( θ s , 0 ) ∗ ν s = μ s , ( θ s , 1 ) ∗ ν s = μ s + 1 , where
θ s , 0 = f ε s − 1 , θ s , 1 = l ε s + 1 − 1 ∘ ( c ↦ a s − 1 c a s ) . \theta_{s,0}=f_{\varepsilon_s}^{-1},\qquad
\theta_{s,1}=l_{\varepsilon_{s+1}}^{-1}\circ(c\mapsto a_s^{-1}ca_s). θ s , 0 = f ε s − 1 , θ s , 1 = l ε s + 1 − 1 ∘ ( c ↦ a s − 1 c a s ) . All the groups K s K_s K s
Proof. Take the sequence from Lemma 3.6, keeping m m m c s ( k ) c_s^{(k)} c s ( k ) L k = ∣ c s ( k ) ∣ P → ∞ L_k=\lvert c_s^{(k)}\rvert_P\to\infty L k = ∣ c s ( k ) ∣ P → ∞ K s K_s K s f ε s ( P ) f_{\varepsilon_s}(P) f ε s ( P ) f ε s ( c s ( k ) ) f_{\varepsilon_s}(c_s^{(k)}) f ε s ( c s ( k ) ) K s K_s K s
Represent f ε s ( c s ( k ) ) f_{\varepsilon_s}(c_s^{(k)}) f ε s ( c s ( k ) ) W s ( k ) W_s^{(k)} W s ( k ) K s K_s K s b s ( k ) b_s^{(k)} b s ( k )
ν s ( k ) = L k − 1 η W s ( k ) b s ( k ) . \nu_s^{(k)}=L_k^{-1}\eta_{W_s^{(k)}b_s^{(k)}}. ν s ( k ) = L k − 1 η W s ( k ) b s ( k ) . The injection θ s , 0 \theta_{s,0} θ s , 0 ∣ W s ( k ) ∣ K s \lvert W_s^{(k)}\rvert_{K_s} ∣ W s ( k ) ∣ K s ∣ c s ( k ) ∣ P \lvert c_s^{(k)}\rvert_P ∣ c s ( k ) ∣ P ∥ ν s ( k ) ∥ \lVert\nu_s^{(k)}\rVert ∥ ν s ( k ) ∥ s s s ν s ( k ) \nu_s^{(k)} ν s ( k ) ν s \nu_s ν s
Apply Lemma 3.9 to both injections from K s K_s K s
θ s , 0 ( W s ( k ) ) = c s ( k ) , θ s , 1 ( W s ( k ) ) = c s + 1 ( k ) . \theta_{s,0}(W_s^{(k)})=c_s^{(k)},\qquad
\theta_{s,1}(W_s^{(k)})=c_{s+1}^{(k)}. θ s , 0 ( W s ( k ) ) = c s ( k ) , θ s , 1 ( W s ( k ) ) = c s + 1 ( k ) . For every fixed word cylinder, the two induced normalized currents therefore differ from the corresponding normalized symmetrized linear row frequencies by O ( 1 ) / L k O(1)/L_k O ( 1 ) / L k c s ( k ) c_s^{(k)} c s ( k ) μ s \mu_s μ s
The length conclusion (3.10) gives
∥ μ s ∥ = lim k ∣ c s ( k ) ∣ P L k \lVert\mu_s\rVert=\lim_{k}\frac{\lvert c_s^{(k)}\rvert_P}{L_k} ∥ μ s ∥ = k lim L k ∣ c s ( k ) ∣ P along this subsequence. The middle ratio is 1 and the two end ratios are less than 2, proving (3.11). The errors used here may depend on m m m K s K_s K s m m m L k → ∞ L_k\to\infty L k → ∞
Finite transition types and a bilateral limit
The preceding construction produces one finite current trajectory for each m m m m m m
For each ordered sign pair ( ϵ , δ ) (\epsilon,\delta) ( ϵ , δ ) d d d
f ϵ ( P ) \ H / l δ ( P ) f_{\epsilon}(P)\backslash H/l_{\delta}(P) f ϵ ( P ) \ H / l δ ( P ) with nontrivial intersection
K ϵ , δ , d = f ϵ ( P ) ∩ d l δ ( P ) d − 1 . K_{\epsilon,\delta,d}=f_{\epsilon}(P)\cap dl_{\delta}(P)d^{-1}. K ϵ , δ , d = f ϵ ( P ) ∩ d l δ ( P ) d − 1 . There are finitely many, by Lemma 2.2 and torsion-freeness of H H H d = 1 d=1 d = 1 τ = ( ϵ , δ , d ) \tau=(\epsilon,\delta,d) τ = ( ϵ , δ , d ) transition types , and write
K τ = K ϵ , δ , d , θ τ , 0 = f ϵ − 1 , θ τ , 1 = l δ − 1 ∘ ( c ↦ d − 1 c d ) . (10) K_{\tau}=K_{\epsilon,\delta,d},\qquad\theta_{\tau,0}=f_{\epsilon}^{-1},\qquad\theta_{\tau,1}=l_{\delta}^{-1}\circ(c\mapsto d^{-1}cd). \tag*{(10)} K τ = K ϵ , δ , d , θ τ , 0 = f ϵ − 1 , θ τ , 1 = l δ − 1 ∘ ( c ↦ d − 1 c d ) . (10) Fix a free basis for every K τ K_{\tau} K τ
Lemma 3.11 (Uniformity of transition types). Every consecutive pair of currents from Lemma 3.10 can be expressed as
μ s = ( θ τ s , 0 ) ∗ ν s , μ s + 1 = ( θ τ s , 1 ) ∗ ν s \mu_s=(\theta_{\tau_s,0})_*\nu_s,\qquad\mu_{s+1}=(\theta_{\tau_s,1})_*\nu_s μ s = ( θ τ s , 0 ) ∗ ν s , μ s + 1 = ( θ τ s , 1 ) ∗ ν s for a current on one of the fixed groups K τ s K_{\tau_s} K τ s ( ϵ s , ϵ s + 1 ) (\epsilon_s,\epsilon_{s+1}) ( ϵ s , ϵ s + 1 ) C ≥ 2 C\ge2 C ≥ 2 D ≥ 1 D\ge1 D ≥ 1 m m m
C − 1 ∥ μ s ∥ ≤ ∥ μ s + 1 ∥ ≤ C ∥ μ s ∥ , ∥ ν s ∥ ≤ D ∥ μ s ∥ . (11) C^{-1}\lVert\mu_s\rVert\le\lVert\mu_{s+1}\rVert\le C\lVert\mu_s\rVert,\qquad\lVert\nu_s\rVert\le D\lVert\mu_s\rVert. \tag*{(11)} C − 1 ∥ μ s ∥ ≤ ∥ μ s + 1 ∥ ≤ C ∥ μ s ∥ , ∥ ν s ∥ ≤ D ∥ μ s ∥ . (11) If K τ K_{\tau} K τ d = 1 d=1 d = 1 i , j i,j i , j j , i j,i j , i
Proof. Suppress the row index and write f = f ϵ f=f_{\epsilon} f = f ϵ l = l δ l=l_{\delta} l = l δ a = f ( p ) d l ( q ) a=f(p)dl(q) a = f ( p ) d l ( q ) p , q ∈ P p,q\in P p , q ∈ P f ( p ) f(p) f ( p ) K d = f ( P ) ∩ d l ( P ) d − 1 K_d=f(P)\cap dl(P)d^{-1} K d = f ( P ) ∩ d l ( P ) d − 1 K a = f ( P ) ∩ a l ( P ) a − 1 K_a=f(P)\cap al(P)a^{-1} K a = f ( P ) ∩ a l ( P ) a − 1 K a K_a K a
k ⟼ p θ τ , 0 ( k ) p − 1 , k ⟼ q − 1 θ τ , 1 ( k ) q . k\longmapsto p\theta_{\tau,0}(k)p^{-1},\qquad k\longmapsto q^{-1}\theta_{\tau,1}(k)q. k ⟼ p θ τ , 0 ( k ) p − 1 , k ⟼ q − 1 θ τ , 1 ( k ) q . By Lemma 3.8, these inner automorphisms leave the induced currents unchanged. Transporting the transition current along the group isomorphism therefore gives the desired representation. Essentiality is unchanged: for a reversal, membership of the bridge in the common side subgroup depends only on its double coset.
Each of the finitely many maps in (10) has upper and positive lower norm bounds from Lemma 3.8. Comparing the two bounds for a type and then taking maxima over the finite type set proves (11) .
If K τ K_{\tau} K τ d d d I I I (10) gives i , j i,j i , j ( + , + , 1 ) (+,+,1) ( + , + , 1 ) j , i j,i j , i ( − , − , 1 ) (-,-,1) ( − , − , 1 )
Proposition 3.12 (The current trajectory forced by nonflaring). Under the hypotheses of Theorem 3.1, if G G G μ s ∈ Curr ( P ) \mu_s \in\operatorname{Curr}(P) μ s ∈ Curr ( P ) s ∈ Z s \in\mathbb{Z} s ∈ Z ϵ s ∈ { + , − } \epsilon_s \in\{+,-\} ϵ s ∈ { + , − } τ s \tau_s τ s ( ϵ s , ϵ s + 1 ) (\epsilon_s,\epsilon_{s+1}) ( ϵ s , ϵ s + 1 )
(i) ∥ μ 0 ∥ = 1 \lVert\mu_0\rVert= 1 ∥ μ 0 ∥ = 1 every μ s \mu_s μ s is nonzero, and sup s ∈ Z ∥ μ s ∥ < ∞ \sup_{s\in\mathbb{Z}}\lVert\mu_s\rVert< \infty sup s ∈ Z ∥ μ s ∥ < ∞
(ii) for every s s s there is a current ν s ∈ Curr ( K τ s ) \nu_s \in\operatorname{Curr}(K_{\tau_s}) ν s ∈ Curr ( K τ s ) with
μ s = ( θ τ s , 0 ) ∗ ν s , μ s + 1 = ( θ τ s , 1 ) ∗ ν s ; \mu_s = (\theta_{\tau_s,0})_*\nu_s,\qquad\mu_{s+1} = (\theta_{\tau_s,1})_*\nu_s; μ s = ( θ τ s , 0 ) ∗ ν s , μ s + 1 = ( θ τ s , 1 ) ∗ ν s ; (iii) all types are essential: any reversing type uses a double coset outside the common side subgroup.
The constants in (11) apply to this trajectory. In particular, ∥ μ s ∥ ≥ C − ∣ s ∣ \lVert\mu_s\rVert\ge C^{-|s|} ∥ μ s ∥ ≥ C − ∣ s ∣ s s s
Proof. For each m m m
D m = max − m ≤ s ≤ m ∥ μ s ∥ . D_m = \max_{-m\le s\le m}\lVert\mu_s\rVert. D m = − m ≤ s ≤ m max ∥ μ s ∥ . The constants C , D C,D C , D (11) are now independent of m m m
Suppose first that D m D_m D m m → ∞ m \to\infty m → ∞ (11) . The sign and type sets are finite. Successively extract subsequences on the windows [ − n , n ] [-n,n] [ − n , n ] s ∈ Z s \in\mathbb{Z} s ∈ Z ∥ μ 0 ∥ = 1 \lVert\mu_0\rVert= 1 ∥ μ 0 ∥ = 1
In the remaining case choose a subsequence with D m → ∞ D_m \to\infty D m → ∞ j m j_m j m D m D_m D m j m j_m j m 0 0 0 1 1 1 1 1 1 2 2 2
D m ≤ 2 C j m + m , D m ≤ 2 C m − j m . D_m \le2C^{j_m+m},\qquad D_m \le2C^{m-j_m}. D m ≤ 2 C j m + m , D m ≤ 2 C m − j m . Thus
min { j m + m , m − j m } ≥ log C ( D m / 2 ) ⟶ ∞ . (12) \min\{j_m+m,m-j_m\} \ge\log_C(D_m/2) \longrightarrow\infty. \tag*{(12)} min { j m + m , m − j m } ≥ log C ( D m /2 ) ⟶ ∞. (12) Every fixed bilateral window is eventually present in the recentered trajectory. The same diagonal extraction applies; the transition norms remain uniformly bounded after normalization. It produces a bilateral trajectory with central norm 1 1 1 1 1 1
In either case, the finite-type equalities and their norm comparisons survive the limit. Applying the lower comparison repeatedly from row 0 0 0 ∥ μ s ∥ ≥ C − ∣ s ∣ \lVert\mu_s\rVert\ge C^{-|s|} ∥ μ s ∥ ≥ C − ∣ s ∣
There are two distinct limits in this proof. At each fixed window length, first send the original annular girth to infinity, with all actual bridges fixed. Only after those errors have disappeared do we replace bridges by finitely many types and let the window length grow. The quantitative estimate (12) ensures that recentering cannot lose either half of the trajectory, while continuity of the norm preserves nonzero mass. We have therefore reduced Theorem 3.1 to excluding the bounded current trajectory of Proposition 3.12 under BS-freeness.
A stationary current and its line trajectories
We retain the restricted HNN splitting and notation of Theorem 3.1. Thus
G = ⟨ H , t ∣ t A t − 1 = B ⟩ , P = A , I = A ∩ B . G=\langle H,t\mid tAt^{-1}=B\rangle,\qquad P=A,\qquad I=A\cap B. G = ⟨ H , t ∣ t A t − 1 = B ⟩ , P = A , I = A ∩ B . and i , j : I ⟶ P i,j:I\longrightarrow P i , j : I ⟶ P t − 1 t^{-1} t − 1 I , J I,J I , J G G G
i ∗ ν = j ∗ ν = μ . i_*\nu=j_*\nu=\mu. i ∗ ν = j ∗ ν = μ . We will then interpret this equality as a measure-preserving transfer of almost every line along a unique path in the Bass–Serre tree T T T
Periodic axes and a finite set of exceptional roots
A root of an axis in the Cayley tree of P P P P P P
Lemma 4.1. Every atom of a free-group current is a periodic axis. If the mass of one oriented axis is b > 0 b>0 b > 0 r r r b ∣ r ∣ c y c b|r|_{\mathrm{cyc}} b ∣ r ∣ cyc
Proof. Translate successive vertices of an atomic line to the identity. All resulting lines have the same positive mass and pass through the identity. That set of lines is compact and has finite measure, so two of the translates coincide. A nontrivial element of the free group therefore preserves the original line. An element reversing a line would fix a vertex or invert an edge, which is impossible for the free action on the Cayley tree. The line stabilizer consequently acts by translations and is infinite cyclic.
For its root r r r ∣ r ∣ c y c |r|_{\mathrm{cyc}} ∣ r ∣ cyc ⟨ r ⟩ \langle r\rangle ⟨ r ⟩
Recall that a transition type τ = ( ϵ , δ , d ) \tau=(\epsilon,\delta,d) τ = ( ϵ , δ , d ) K τ K_\tau K τ
θ τ , 0 , θ τ , 1 : K τ ⟶ P . \theta_{\tau,0},\theta_{\tau,1}:K_\tau\longrightarrow P. θ τ , 0 , θ τ , 1 : K τ ⟶ P . There are finitely many types in Proposition 3.12. Their traversal signs and double-coset representatives are part of the type. A noncyclic type must be aligned: its two signs agree, its representative is 1 1 1 i , j i,j i , j
Lemma 4.2. There is a finite set D D D
(i) The root of either image of a nontrivial element of a cyclic transition group belongs to D D D
(ii) If a periodic axis is carried by p A E pA_E p A E q A E qA_E q A E p E ≠ q E pE\ne qE pE = q E E = I E=I E = I E = J E=J E = J D D D
(iii) If an axis root in a noncyclic transition group maps to a proper power under either transition injection, the root of that image belongs to D D D
Here “carried” means that both endpoints lie in the indicated translate of the limit set.
Proof. For each nontrivial cyclic transition group, include the roots of the images of one generator under both injections. This makes a finite set and proves (i).
For E = I , J E=I,J E = I , J E E E E ∩ x E x − 1 E\cap xEx^{-1} E ∩ x E x − 1 x ∉ E x\notin E x ∈ / E r ˉ ( E ) \bar r(E) r ˉ ( E ) E x E ExE E x E P P P
Suppose that an axis is carried by p Λ E p\Lambda E p Λ E q Λ E q\Lambda E q Λ E p E ≠ q E pE\ne qE pE = q E p − 1 p^{-1} p − 1
Λ E ∩ Λ ( ( p − 1 q ) E ( p − 1 q ) − 1 ) = Λ ( E ∩ ( p − 1 q ) E ( p − 1 q ) − 1 ) . \Lambda E\cap\Lambda\left((p^{-1}q)E(p^{-1}q)^{-1}\right)=\Lambda\left(E\cap(p^{-1}q)E(p^{-1}q)^{-1}\right). Λ E ∩ Λ ( ( p − 1 q ) E ( p − 1 q ) − 1 ) = Λ ( E ∩ ( p − 1 q ) E ( p − 1 q ) − 1 ) . The intersection subgroup is nontrivial cyclic. If p − 1 q = e 1 x e 2 p^{-1}q=e_{1}xe_{2} p − 1 q = e 1 x e 2 e 1 , e 2 ∈ E e_{1},e_{2}\in E e 1 , e 2 ∈ E x x x e 1 e_{1} e 1 E ∩ x E x − 1 E\cap xEx^{-1} E ∩ x E x − 1
Finally, suppose an axis root u u u r k r^{k} r k r r r P P P ∣ k ∣ > 1 |k|>1 ∣ k ∣ > 1 E E E I I I J J J r ∉ E r\notin E r ∈ / E r r r u u u E , r E E,rE E , r E r r r
We next record exactly how a repetition in this finite set produces a group-theoretic obstruction. The observation applies even when the traversal signs change between the two repeated positions.
Lemma 4.3. Let ( h s e ) (h_s e) ( h s e ) T \mathcal{T} T r s ∈ P r_s\in P r s ∈ P
h s ⟨ r s ⟩ h s − 1 h_s\langle r_s\rangle h_s^{-1} h s ⟨ r s ⟩ h s − 1 at consecutive positions are commensurable. If i < j i<j i < j ⟨ r i ⟩ , ⟨ r j ⟩ \langle r_i\rangle,\langle r_j\rangle ⟨ r i ⟩ , ⟨ r j ⟩ P P P G G G
Proof. Choose p ∈ P p\in P p ∈ P p ⟨ r i ⟩ p − 1 = ⟨ r j ⟩ p\langle r_i\rangle p^{-1}=\langle r_j\rangle p ⟨ r i ⟩ p − 1 = ⟨ r j ⟩ g = h j p h i − 1 g=h_jph_i^{-1} g = h j p h i − 1 p p p e e e h i e h_i e h i e h j e h_j e h j e i i i j j j
Let σ \sigma σ h i e h_i e h i e h j e h_j e h j e g k σ g^k\sigma g k σ g g g g g g
The cyclic subgroup at position i i i g g g
Proposition 4.4. Let ( μ s ) s ∈ Z (\mu_s)_{s\in\mathbb{Z}} ( μ s ) s ∈ Z P P P G G G μ s \mu_s μ s Proof. Suppose a row has an atom, and start indexing there. At each forward transition, the coset formula for the induced current has a positive contribution to this atom from an atom of the transition current. Choose one such atom and take its image in the next row. Local finiteness of the coset sum and injectivity of the boundary maps in Lemma 3.8 justify this choice. Lemma 4.1 makes every chosen line a periodic axis. Let r s r_s r s s s s
We first develop these matches into an actual reduced tree path, so that the repetition criterion of Lemma 4.3 will apply. For a transition with representative d d d f , l : P → H f,l:P\to H f , l : P → H k ∈ K τ k\in K_\tau k ∈ K τ
θ τ , 0 ( k ) = p − 1 r s a p , θ τ , 1 ( k ) = q − 1 r s + 1 b q , a , b ∈ Z ∖ { 0 } . \theta_{\tau,0}(k)=p^{-1}r_s^a p,\qquad\theta_{\tau,1}(k)=q^{-1}r_{s+1}^b q,\qquad a,b\in\mathbb{Z}\setminus\{0\}. θ τ , 0 ( k ) = p − 1 r s a p , θ τ , 1 ( k ) = q − 1 r s + 1 b q , a , b ∈ Z ∖ { 0 } . The bridge
f ( p ) d l ( q ) − 1 f(p)d l(q)^{-1} f ( p ) d l ( q ) − 1 therefore matches r s a r_s^a r s a r s + 1 b r_{s+1}^b r s + 1 b d d d (3.4) . They give a reduced path with the prescribed signs, and the framed cyclic subgroups are commensurable at every step. This construction works for arbitrary nonzero exponents, including negative ones. It therefore suffices to find two positions with the same root class and traversal sign.
Call a transition exceptional for this chosen trajectory if its group is cyclic, if either chosen image line is carried by more than one coset of its image subgroup, or if an upstairs root has a proper-power image. By Lemma 4.2, at least one of the adjacent roots at every exceptional transition belongs to D D D
If there are only finitely many exceptional transitions, discard an initial segment. On the remaining tail there is exactly one contributing coset on each side of every push formula. Hence both image atoms have exactly the mass of their common upstairs atom. The chosen atoms have one fixed positive mass on this tail. The norm bound and Lemma 4.1 give a uniform bound on ∣ r s ∣ c y c |r_s|_{\mathrm{cyc}} ∣ r s ∣ cyc
In either case, Lemma 4.3 gives a Baumslag–Solitar subgroup, a contradiction. □
A stationary current
We now have an atomless bounded trajectory. The next argument turns it into a stationary one; the case in which the total mass is summable must be treated separately from averaging.
Proposition 4.5. Under the assumptions of Proposition 4.4, there exist a nonzero current ν \nu ν I I I μ \mu μ P P P
i ∗ ν = j ∗ ν = μ . (13) i_*\nu=j_*\nu=\mu. \tag*{(13)} i ∗ ν = j ∗ ν = μ . (13) In particular, if the restricted HNN extension is not hyperbolic, Proposition 3.12 supplies such a stationary current.
Proof. A nonzero current on an infinite cyclic group induces an atomic current. Thus no transition in the atomless trajectory is cyclic. Every transition is consequently aligned, all traversal signs are the same, and its maps are i , j i,j i , j j , i j,i j , i ν s \nu_s ν s I I I
i ∗ ν s = μ s , j ∗ ν s = μ s + 1 . i_*\nu_s=\mu_s,\qquad j_*\nu_s=\mu_{s+1}. i ∗ ν s = μ s , j ∗ ν s = μ s + 1 . In particular I I I I I I
S N = ∑ s = − N N ν s , a N = ∥ S N ∥ = ∑ s = − N N ∥ ν s ∥ . S_N=\sum_{s=-N}^{N}\nu_s,\qquad a_N=\lVert S_N\rVert=\sum_{s=-N}^{N}\lVert\nu_s\rVert. S N = s = − N ∑ N ν s , a N = ∥ S N ∥ = s = − N ∑ N ∥ ν s ∥ . The last equality uses positivity. Telescoping gives
j ∗ S N − i ∗ S N = μ N + 1 − μ − N . j_*S_N-i_*S_N=\mu_{N+1}-\mu_{-N}. j ∗ S N − i ∗ S N = μ N + 1 − μ − N . First suppose a N → ∞ a_N\to\infty a N → ∞ S N / a N S_N/a_N S N / a N ν \nu ν (4.2) , divided by a N a_N a N i ∗ ν = j ∗ ν i_*\nu=j_*\nu i ∗ ν = j ∗ ν μ \mu μ
Now suppose ( a N ) (a_N) ( a N )
ν = ∑ s ∈ Z ν s \nu=\sum_{s\in\mathbb{Z}}\nu_s ν = s ∈ Z ∑ ν s has finite positive norm and defines a Radon measure: every compact line set is covered by finitely many translates of the identity line set. Monotone convergence permits induction through this sum. Reindexing the resulting positive series gives
i ∗ ν = ∑ s ∈ Z μ s = ∑ s ∈ Z μ s + 1 = j ∗ ν . i_*\nu=\sum_{s\in\mathbb{Z}}\mu_s=\sum_{s\in\mathbb{Z}}\mu_{s+1}=j_*\nu. i ∗ ν = s ∈ Z ∑ μ s = s ∈ Z ∑ μ s + 1 = j ∗ ν . These currents have finite norm by the upper norm bound for induction, and are nonzero. Equivalently, summability and norm comparison make the endpoint norms in (4.2) tend to zero.
In either case we must still prove atomlessness: a weak limit of atomless currents can have atoms. Use the constant bilateral trajectory with row μ \mu μ ν \nu ν μ \mu μ ν \nu ν ν \nu ν
Transfers on a conull set of lines
Fix the currents in (4.1), and let Ω = supp μ ⊂ ∂ 2 P \Omega=\operatorname{supp}\mu\subset\partial^2P Ω = supp μ ⊂ ∂ 2 P rational if a nontrivial element of the free group fixes it. Equivalently, its ray labels are eventually periodic. We first remove all lines with a rational endpoint.
Lemma 4.6. For an atomless free-group current, almost every line has two irrational endpoints. An injection between finitely generated free groups preserves and reflects rationality of boundary points. Proof. Fix a rational endpoint ξ \xi ξ ξ \xi ξ v v v D v D_v D v r r r ξ \xi ξ r − n D v r^{-n}D_v r − n D v n ≥ 0 n \ge0 n ≥ 0 v v v r − n v r^{-n}v r − n v v v v μ ( D v ) = 0 \mu(D_v)=0 μ ( D v ) = 0
An injection preserves rationality by equivariance of its boundary map. To prove reflection, identify its image with a finitely generated subgroup E E E ξ ∈ Λ E \xi\in\Lambda E ξ ∈ Λ E r r r r r r ξ \xi ξ x x x r n x r^n x r n x E E E ξ ∈ Λ E \xi\in\Lambda E ξ ∈ Λ E e n ∈ E e_n \in E e n ∈ E r n x r^n x r n x ( r n x ) − 1 e n (r^n x)^{-1}e_n ( r n x ) − 1 e n n > m n>m n > m
e n e m − 1 = r n − m ∈ E . e_n e_m^{-1}=r^{n-m}\in E. e n e m − 1 = r n − m ∈ E . The inverse image of this nontrivial element fixes the original boundary point, proving reflection. □
We write ∂ 2 Λ E \partial^2\Lambda E ∂ 2 Λ E Λ E \Lambda E Λ E E = I E=I E = I J J J p Λ E p\Lambda E p Λ E p E pE pE
Proposition 4.7. There is a translation- and flip-invariant conull Borel set U ⊂ Ω U\subset\Omega U ⊂ Ω
(i) Every line of U U U I I I J J J
(ii) On lines carried by p Λ I p\Lambda I p Λ I p − 1 p^{-1} p − 1 I I I J J J q Λ J q\Lambda J q Λ J μ \mu μ
(iii) Every defined transfer, its inverse, and every output frame translation take lines of U U U U U U U U U
Measure preservation in (ii) holds on the full Borel domain of lines with irrational endpoints, not only after restriction to U U U
Proof. Let X ⊂ ∂ 2 P X\subset\partial^2P X ⊂ ∂ 2 P i ∗ v i_*v i ∗ v j ∗ v j_*v j ∗ v X X X I I I J J J
Let ϑ : ∂ I → ∂ J \vartheta:\partial I\to\partial J ϑ : ∂ I → ∂ J j j j I I I i i i
D p = X ∩ ∂ 2 ( p Λ I ) D_p=X\cap\partial^2(p\Lambda I) D p = X ∩ ∂ 2 ( p Λ I ) define F p ( L ) = ϑ ( p − 1 L ) F_p(L)=\vartheta(p^{-1}L) F p ( L ) = ϑ ( p − 1 L ) F p ( D p ) = X ∩ ∂ 2 Λ J F_p(D_p)=X\cap\partial^2\Lambda J F p ( D p ) = X ∩ ∂ 2 Λ J D ⊂ D p D\subset D_p D ⊂ D p
μ ( D ) = ν ( p − 1 D ) = μ ( F p D ) . \mu(D)=\nu(p^{-1}D)=\mu(F_pD). μ ( D ) = ν ( p − 1 D ) = μ ( F p D ) . The middle term is expressed in the boundary coordinates of I I I
For completeness, we now choose a single conull set on which all these operations can be iterated. Start in X X X
B = ( X ∖ Ω ) ∪ { L ∈ X : L has no I -coset carrier } ∪ { L ∈ X : L has no J -coset carrier } . B = (X \setminus\Omega)
\cup\{L \in X : L\text{ has no }I\text{-coset carrier}\}
\cup\{L \in X : L\text{ has no }J\text{-coset carrier}\}. B = ( X ∖ Ω ) ∪ { L ∈ X : L has no I -coset carrier } ∪ { L ∈ X : L has no J -coset carrier } . The complement of the support is null because line space has a countable base. Saturate B B B (4.3) shows that each composition takes null sets to null sets on its domain. Their saturation is therefore Borel and null. Its complement in X X X U \mathcal{U} U X ∖ Ω X \setminus\Omega X ∖ Ω
The set U \mathcal{U} U Ω \Omega Ω U \mathcal{U} U
Actual paths and finite windows
We next describe precisely which tree path is followed by a line in U \mathcal{U} U
Proposition 4.8. Every line L ∈ U L \in\mathcal{U} L ∈ U
⋯ , E − 1 , E 0 = e , E 1 , ⋯ \cdots, E_{-1}, E_0 = e, E_1, \cdots ⋯ , E − 1 , E 0 = e , E 1 , ⋯ in T T T
(i) The path is independent of coset representatives and of compatible changes of frames and line coordinates.
(ii) For every finite window containing E 0 E_0 E 0 K ≤ P K \le P K ≤ P L L L Λ K \Lambda_K Λ K
(iii) Two distinct actual aligned windows of the same shape about e e e
(iv) If an aligned one-sided window joins e e e h e he h e S = P ∩ h P h − 1 S = P \cap hPh^{-1} S = P ∩ h P h − 1
S ⟶ h − 1 S h , x ⟼ h − 1 x h , S \longrightarrow h^{-1}Sh,\qquad x \longmapsto h^{-1}xh, S ⟶ h − 1 S h , x ⟼ h − 1 x h , implements its line transfer. On the irrational-endpoint domain it preserves μ \mu μ U \mathcal{U} U U \mathcal{U} U
In (iii), a shape specifies the numbers of edges before and after the central edge, all traversed positively.
Proof. In a frame h h h p I pI p I h p t e hpt e h pt e p J pJ p J h p t − 1 e hpt^{-1}e h p t − 1 e
Changing p p p p i 0 pi_0 p i 0 i 0 ∈ I i_0 \in I i 0 ∈ I
p i 0 t = p t j ( i 0 ) . pi_0t = pt\,j(i_0). p i 0 t = pt j ( i 0 ) . It changes its frame on the right by j ( i 0 ) ∈ P j(i_0) \in P j ( i 0 ) ∈ P J J J P P P h p t hpt h pt ∂ 2 Λ J \partial^2\Lambda_J ∂ 2 Λ J J J J h p t t − 1 e = h p e = h e hptt^{-1}e = hpe = he h pt t − 1 e = h p e = h e h p hp h p
To prove (ii), build a finite window one end at a time. Initially its stabilizer is P P P K ′ K' K ′ p I p − 1 pIp^{-1} p I p − 1 p J p − 1 pJp^{-1} p J p − 1
For (iii), Lemma 3.4 gives the cyclic common stabilizer at the first divergence in either direction. The window stabilizers are finitely generated, so their limit sets intersect in the limit set of their intersection. This proves the assertion about irrational endpoints as well. The same proof compares restrictions of longer paths to any common finite shape.
An element fixing the two endpoint edges of a tree segment fixes every edge between them. Thus the stabilizer in (iv) is precisely P ∩ h P h − 1 P \cap hPh^{-1} P ∩ h P h − 1 h − 1 h^{-1} h − 1 Λ S \Lambda S Λ S U \mathcal{U} U
Lemma 4.9. Fix a finite positive window shape about e e e P P P
Proof. The window bound, Lemma 3.3, gives finitely many P P P W 1 , … , W k W_1,\ldots,W_k W 1 , … , W k K 1 , … , K k K_1,\ldots,K_k K 1 , … , K k i i i Hull ( Λ K i ) \operatorname{Hull}(\Lambda K_i) Hull ( Λ K i ) K i K_i K i K i K_i K i v i , j v_{i,j} v i , j p W i pW_i p W i x x x p − 1 x = k v i , j p^{-1}x = kv_{i,j} p − 1 x = k v i , j k ∈ K i k \in K_i k ∈ K i j j j
p W i = x v i , j − 1 W i . pW_i = xv_{i,j}^{-1}W_i. p W i = x v i , j − 1 W i . because k k k W i W_i W i x , i , j x,i,j x , i , j
We have obtained a nonzero stationary current and a conull family of unique bilateral trajectories. Every finite part of such a trajectory has a noncyclic stabilizer of the uniformly bounded total reduced rank from Lemma 3.3. The next section encodes these stabilizers by finite core graphs. Passing to the closed support there will use closedness of finite-graph liftability, rather than treating a conull assertion as a support assertion.
Finite carriers for a stationary current
We continue with the restricted HNN splitting, the free group P P P
i ∗ ν = j ∗ ν = μ i_*\nu= j_*\nu= \mu i ∗ ν = j ∗ ν = μ provided by Proposition 4.5. Let T P T_P T P P P P R R R Ω = supp μ \Omega= \operatorname{supp}\mu Ω = supp μ U ⊂ Ω U \subset\Omega U ⊂ Ω e e e T T T
Our objective is to identify the finitely generated noncyclic subgroups that can fix an entire bilateral trajectory. The window stabilizers have bounded rank but can have arbitrarily large core graphs. We keep track of that distinction: after suppressing vertices of degree two, the graph sizes are bounded, and the edges whose labels remain bounded produce a finite family of subgroup carriers. Maximal carriers return to themselves under transfer, giving automorphisms to which we can apply Brinkmann’s theorem.
Window cores and their limits
We use the immersed-core description of finitely generated free subgroups and the corresponding pullback description of intersections; see Stallings [22 ] , Section 3.3, Theorem 5.5, and Section 7. The rank bounds and limiting carrier construction needed here are proved below.
For a finitely generated nontrivial subgroup K ≤ P K \le P K ≤ P
C ( K ) = K \ Hull T P ( Λ K ) . C(K) = K \backslash\operatorname{Hull}_{T_P}(\Lambda_K). C ( K ) = K \ Hull T P ( Λ K ) . This is its finite unbased core graph, with its label-preserving immersion into R R R
For each n ≥ 1 n \ge1 n ≥ 1 P P P
W = ( E − n , … , E n ) , E 0 = e , W = (E_{-n},\ldots,E_n), \qquad E_0 = e, W = ( E − n , … , E n ) , E 0 = e , whose full pointwise stabilizer K W = Stab P ( W ) K_W = \operatorname{Stab}_P(W) K W = Stab P ( W )
Γ n = ∐ W C ( K W ) , Γ n ⟶ R . \Gamma_n = \coprod_W C(K_W), \qquad\Gamma_n \longrightarrow R. Γ n = W ∐ C ( K W ) , Γ n ⟶ R . The choice is finite by Lemma [3 ] . Set r 0 = r ˉ ( P ) r_0=\bar r(P) r 0 = r ˉ ( P ) μ \mu μ P P P r 0 ≥ 1 r_0\ge1 r 0 ≥ 1
A state of a labeled core is one of its vertices. In each noncyclic core, call a vertex natural if its degree is at least three. A natural edge is a maximal nonbacktracking path whose interior vertices have degree two. Suppressing those interiors gives the natural graph .
Lemma 5.1 (Window cores). The following properties hold.
The sum of the reduced ranks of the components of Γ n \Gamma_n Γ n r 0 r_0 r 0 Γ n \Gamma_n Γ n 3 r 0 3r_0 3 r 0 2 r 0 2r_0 2 r 0
Every line in Ω \Omega Ω Γ n \Gamma_n Γ n
A ray with irrational endpoint has at most one starting state for a lift among all components of Γ n \Gamma_n Γ n
The same statements hold for any other fixed finite positive window shape containing its specified central edge, with the same bounds.
Proof. The reduced-rank assertion is Lemma [3 ] . In a noncyclic connected core, suppressing degree-two interiors leaves a graph all of whose vertices have degree at least three. If E , V E,V E , V r r r
2 E ≥ 3 V , E − V = r − 1. 2E \ge3V,\qquad E-V=r-1. 2 E ≥ 3 V , E − V = r − 1. Consequently E ≤ 3 ( r − 1 ) E\le3(r-1) E ≤ 3 ( r − 1 ) V ≤ 2 ( r − 1 ) V\le2(r-1) V ≤ 2 ( r − 1 )
We describe precisely what a state of a core represents. If the representative window is W W W K = K W K=K_W K = K W K x Kx K x C ( K ) C(K) C ( K ) x − 1 W x^{-1}W x − 1 W e e e x − 1 W = y − 1 W x^{-1}W=y^{-1}W x − 1 W = y − 1 W y x − 1 ∈ Stab P ( W ) = K yx^{-1}\in\operatorname{Stab}_P(W)=K y x − 1 ∈ Stab P ( W ) = K K x = K y Kx=Ky K x = K y P P P
A ray starting at K x Kx K x T P T_P T P x x x Hull ( Λ K ) \operatorname{Hull}(\Lambda_K) Hull ( Λ K ) Λ ( x − 1 K x ) \Lambda(x^{-1}Kx) Λ ( x − 1 K x ) [4 ] forbids two distinct represented windows carrying that endpoint. This proves uniqueness of the starting state. Continuation from a state is unique because the core map is an immersion.
For a line in U \mathcal{U} U [4 ] ; hence its labels lift to Γ n \Gamma_n Γ n U \mathcal{U} U Ω \Omega Ω Ω \Omega Ω
The proof applies to any fixed window shape, using the corresponding rank bound and trajectory uniqueness. □
We shall also use Lemma [9 ] : for a fixed window shape and a fixed ball in T P T_P T P x − 1 Hull ( Λ K ) x^{-1}\operatorname{Hull}(\Lambda_K) x − 1 Hull ( Λ K ) y y y K x y Kxy K x y C ( K ) C(K) C ( K ) y y y K x y Kxy K x y K x Kx K x x − 1 W x^{-1}W x − 1 W
Definition 5.2. Pass to a subsequence n → ∞ n \to\infty n → ∞ Γ n \Gamma_n Γ n small graph Γ s m \Gamma^{\mathrm{sm}} Γ sm
Here is the order of these choices. There are only finitely many natural graph types with at most 3 r 0 3r_0 3 r 0 2 r 0 2r_0 2 r 0 n → ∞ n \to\infty n → ∞
Figure 1. A fixed natural graph can have arbitrarily long edge labels. Here the bounded loops survive and the divergent bridge is deleted, while both natural vertices remain. The small components in this schematic example are cyclic; a noncyclic small component is not assumed to exist.
Each connected component D D D Γ s m \Gamma^{\mathrm{sm}} Γ sm P P P R \mathbb{R} R π 1 D \pi_1D π 1 D D D D
A finite cofinal family of subgroups
Let C \mathcal{C} C P P P Q ≤ P Q \leq P Q ≤ P e e e
[ Q ] ⪯ [ M ] ⟺ Q ≤ a M a − 1 for some a ∈ P . [Q] \preceq[M] \quad\Longleftrightarrow\quad Q \leq aMa^{-1}\text{ for some }a \in P. [ Q ] ⪯ [ M ] ⟺ Q ≤ a M a − 1 for some a ∈ P . We first verify that this really is a partial order, rather than merely a preorder.
Lemma 5.3. A nontrivial finitely generated subgroup of a finite-rank free group cannot be properly contained in a conjugate of itself. Consequently ⪯ \preceq ⪯ C \mathcal{C} C
Proof. Suppose a − 1 M a ⪯ M a^{-1}Ma \preceq M a − 1 M a ⪯ M a k a^k a k M M M
M ⊇ a − 1 M a ⊇ ⋯ ⊇ a − k M a k = M M \supseteq a^{-1}Ma \supseteq\cdots\supseteq a^{-k}Ma^k = M M ⊇ a − 1 M a ⊇ ⋯ ⊇ a − k M a k = M force equality at every stage. Thus the Schreier vertices M a − n Ma^{-n} M a − n n ≥ 0 n \ge0 n ≥ 0 w ∈ M w \in M w ∈ M L L L w w w a − n w a n ∈ M a^{-n}wa^n \in M a − n w a n ∈ M
The Schreier graph consists of its finite unbased core with trees attached. A closed reduced path of length L L L L / 2 L/2 L /2 L / 2 L/2 L /2 M a − n Ma^{-n} M a − n a a a a − 1 a^{-1} a − 1
If [ Q ] ⪯ [ M ] ⪯ [ Q ] [Q] \preceq[M] \preceq[Q] [ Q ] ⪯ [ M ] ⪯ [ Q ]
Q ≤ a M a − 1 ≤ a b Q ( a b ) − 1 . Q \leq aMa^{-1} \leq abQ(ab)^{-1}. Q ≤ a M a − 1 ≤ ab Q ( ab ) − 1 . The inclusion between the first and last groups is equality by the first assertion. Both intermediate inclusions are therefore equalities, and [ Q ] = [ M ] [Q] = [M] [ Q ] = [ M ]
The following elementary graph bound supplies the compactness step. It is important here that a core can have bridges.
Lemma 5.4 (Bounded circuits in a core). Let C C C C C C 2 ∣ E ( C ) ∣ 2|E(C)| 2∣ E ( C ) ∣ ∣ E ( C ) ∣ |E(C)| ∣ E ( C ) ∣
Proof. A nonbridge edge lies on a simple cycle of length at most ∣ E ( C ) ∣ |E(C)| ∣ E ( C ) ∣ a a a a a a a a a a − 1 a^{-1} a − 1 a a a a a a 2 ∣ E ( C ) ∣ 2|E(C)| 2∣ E ( C ) ∣
Proposition 5.5 (Finite cofinal carriers). The noncyclic component subgroup classes of Γ s m \Gamma^{\mathrm{sm}} Γ sm C \mathcal{C} C [ Q ] ∈ C [Q] \in\mathcal{C} [ Q ] ∈ C P P P C \mathcal{C} C
Proof. Let M M M Γ n \Gamma_n Γ n M M M n n n n n n M M M M M M [ M ] ∈ C [M] \in\mathcal{C} [ M ] ∈ C
Now take [ Q ] ∈ C [Q] \in\mathcal{C} [ Q ] ∈ C C ( Q ) C(Q) C ( Q )
D Q = 2 ∣ E ( C ( Q ) ) ∣ . D_Q = 2|E(C(Q))|. D Q = 2∣ E ( C ( Q )) ∣. For every selected n n n Q Q Q n n n
C ( Q ) ⟶ Γ n . (14) C(Q) \longrightarrow\Gamma_n. \tag*{(14)} C ( Q ) ⟶ Γ n . (14) One way to see the map directly is to include the minimal Q Q Q
Choose n n n Γ n \Gamma_n Γ n D Q D_Q D Q C ( Q ) C(Q) C ( Q ) D Q D_Q D Q D Q D_Q D Q
The whole image of C ( Q ) C(Q) C ( Q ) Γ s m \Gamma^{\mathrm{sm}} Γ sm M M M [ Q ] ⪯ [ M ] [Q] \preceq[M] [ Q ] ⪯ [ M ] M M M n n n D Q D_Q D Q
There are finitely many small components. A maximal member of their noncyclic subgroup classes is maximal in all of C C C C C C
We have now replaced the possibly unbounded window cores by a finite set of maximal subgroup carriers. The next step uses maximality to make a return transfer surjective. If there are no noncyclic small components, C C C
Return automorphisms
Moving one edge forward along the path fixed by Q Q Q Q Q Q
F : C ⟶ C . F:C\longrightarrow C. F : C ⟶ C . This moves subgroup coordinates, not individual support lines; it is defined for every class in C C C
Proposition 5.6 (Atoroidal return). The map F F F C C C M M M p ≥ 1 p \ge1 p ≥ 1 v ∈ G v \in G v ∈ G p p p
v − 1 M v = M . v^{-1}Mv=M. v − 1 M v = M . The automorphism α : M → M \alpha:M\to M α : M → M α ( m ) = v − 1 m v \alpha(m)=v^{-1}mv α ( m ) = v − 1 m v M M M α \alpha α
Proof. Choose a frame h h h Q Q Q h − 1 Q h ⪯ P h^{-1}Qh \preceq P h − 1 Q h ⪯ P h q hq h q q ∈ P q \in P q ∈ P P P P Q Q Q P P P F F F Q ⪯ M Q \preceq M Q ⪯ M
An order automorphism permutes maximal elements. There are finitely many of them by Proposition 5.5, so F p [ M ] = [ M ] F^p[M]=[M] F p [ M ] = [ M ] p ≥ 1 p \ge1 p ≥ 1 N N N p ≤ N p \le N p ≤ N h h h p p p c ∈ P c \in P c ∈ P
h − 1 M h = c M c − 1 . h^{-1}Mh=cMc^{-1}. h − 1 M h = c M c − 1 . Set v = h c v=hc v = h c v v v p p p v − 1 M v = M v^{-1}Mv=M v − 1 M v = M
Suppose α k ( w ) = u − 1 w u \alpha^k(w)=u^{-1}wu α k ( w ) = u − 1 w u w ∈ M w \in M w ∈ M u ∈ M u \in M u ∈ M k ≥ 1 k \ge1 k ≥ 1
g = v k u − 1 g=v^ku^{-1} g = v k u − 1 commutes with w w w k p ≠ 0 kp \ne0 k p = 0 w w w w a g b = 1 w^ag^b=1 w a g b = 1 b = 0 b=0 b = 0 a = 0 a=0 a = 0 ⟨ w , g ⟩ ≅ Z 2 = B S ( 1 , 1 ) \langle w,g\rangle\cong\mathbb{Z}^2=BS(1,1) ⟨ w , g ⟩ ≅ Z 2 = B S ( 1 , 1 ) G G G
The element v v v M M M M M M M M M v e ve v e p p p e e e v b e v^be v b e v − b e v^{-b}e v − b e
Expansion and vanishing carrier mass
For an element w w w ∣ w ∣ |w| ∣ w ∣ ℓ ( w ) \ell(w) ℓ ( w ) α \alpha α b 0 ≥ 1 b_0 \ge1 b 0 ≥ 1 λ 0 > 1 \lambda_0 > 1 λ 0 > 1
λ 0 ∣ w ∣ ≤ max { ∣ α b 0 ( w ) ∣ , ∣ α − b 0 ( w ) ∣ } for every w \lambda_0|w| \le\max\{|\alpha^{b_0}(w)|,|\alpha^{-b_0}(w)|\}\quad\text{for every }w λ 0 ∣ w ∣ ≤ max { ∣ α b 0 ( w ) ∣ , ∣ α − b 0 ( w ) ∣ } for every w [3 ] . These are based lengths in a fixed free basis. We need both a stronger expansion factor and a consequence for arbitrary currents; the latter uses a sum, not a maximum.
Lemma 5.7 (Expansion on elements and currents). Let M M M v v v α \alpha α M M M b ≥ 1 b \ge1 b ≥ 1 β = α b \beta=\alpha^b β = α b λ > 4 \lambda>4 λ > 4
λ ∣ w ∣ M ≤ max { ∣ β ( w ) ∣ M , ∣ β − 1 ( w ) ∣ M } , w ∈ M , (15) \lambda|w|_M \le\max\{|\beta(w)|_M,|\beta^{-1}(w)|_M\},\qquad w\in M, \tag*{(15)} λ ∣ w ∣ M ≤ max { ∣ β ( w ) ∣ M , ∣ β − 1 ( w ) ∣ M } , w ∈ M , (15) λ ℓ M ( w ) ≤ max { ℓ M ( β ( w ) ) , ℓ M ( β − 1 ( w ) ) } , w ∈ M . (16) \lambda\ell_M(w) \le\max\{\ell_M(\beta(w)),\ell_M(\beta^{-1}(w))\},\qquad w\in M. \tag*{(16)} λ ℓ M ( w ) ≤ max { ℓ M ( β ( w )) , ℓ M ( β − 1 ( w ))} , w ∈ M . (16) For every M M M σ \sigma σ
λ ∥ σ ∥ M ≤ ∥ β ∗ σ ∥ M + ∥ ( β − 1 ) ∗ σ ∥ M . (17) \lambda\|\sigma\|_M \le\|\beta_*\sigma\|_M+\|(\beta^{-1})_*\sigma\|_M. \tag*{(17)} λ ∥ σ ∥ M ≤ ∥ β ∗ σ ∥ M + ∥ ( β − 1 ) ∗ σ ∥ M . (17) The partial conjugations along the paths to v b e v^be v b e v − b e v^{-b}e v − b e
S + = P ∩ v b P v − b , S − = P ∩ v − b P v b , S_+=P\cap v^bPv^{-b},\qquad S_-=P\cap v^{-b}Pv^b, S + = P ∩ v b P v − b , S − = P ∩ v − b P v b , respectively. These are finitely generated full window stabilizers, and both contain M M M ∂ 2 Λ S + \partial^2\Lambda S_+ ∂ 2 Λ S + ∂ 2 Λ S − \partial^2\Lambda S_- ∂ 2 Λ S − Θ + , Θ − \Theta_+,\Theta_- Θ + , Θ − ∂ 2 Λ M \partial^2\Lambda M ∂ 2 Λ M β , β − 1 \beta,\beta^{-1} β , β − 1 μ \mu μ
Proof. Apply Brinkmann’s theorem to α \alpha α γ = α b 0 \gamma=\alpha^{b_0} γ = α b 0 ∣ γ ( w ) ∣ M ≥ λ 0 ∣ w ∣ M |\gamma(w)|_M\geq\lambda_0|w|_M ∣ γ ( w ) ∣ M ≥ λ 0 ∣ w ∣ M γ ( w ) \gamma(w) γ ( w ) ∣ w ∣ M |w|_M ∣ w ∣ M λ 0 ∣ γ ( w ) ∣ M \lambda_0|\gamma(w)|_M λ 0 ∣ γ ( w ) ∣ M w ≠ 1 w\ne1 w = 1
∣ γ k ( w ) ∣ M ≥ λ 0 k ∣ w ∣ M . |\gamma^k(w)|_M\geq\lambda_0^k|w|_M. ∣ γ k ( w ) ∣ M ≥ λ 0 k ∣ w ∣ M . The analogous statement holds for a backward initial expanding direction. Choose k ≥ 1 k\geq1 k ≥ 1 λ 0 k > 4 \lambda_0^k>4 λ 0 k > 4 b = b 0 k b=b_0k b = b 0 k β = α b \beta=\alpha^b β = α b λ = λ 0 k \lambda=\lambda_0^k λ = λ 0 k w = 1 w=1 w = 1 w q w^{q} w q q q q q → ∞ q\to\infty q → ∞ lim q → ∞ ∣ w q ∣ M / q = ℓ M ( w ) \lim_{q\to\infty}|w^q|_M/q=\ell_M(w) lim q → ∞ ∣ w q ∣ M / q = ℓ M ( w )
For a counting current η w \eta_w η w ∥ η w ∥ M = ℓ M ( w ) \|\eta_w\|_M=\ell_M(w) ∥ η w ∥ M = ℓ M ( w )
Fix k ≥ 2 k\geq2 k ≥ 2 k − 1 k-1 k − 1 k k k ⟨ u , σ ⟩ \langle u,\sigma\rangle ⟨ u , σ ⟩ k k k u u u ( k − 1 ) (k-1) ( k − 1 )
Write the circulation as ∑ i a i C i \sum_i a_iC_i ∑ i a i C i w i w_i w i C i f l i p C_i^{\mathrm{flip}} C i flip η w i \eta_{w_i} η w i C i + C i f l i p C_i+C_i^{\mathrm{flip}} C i + C i flip σ \sigma σ
σ k = 1 2 ∑ i a i η w i \sigma_k=\frac{1}{2}\sum_i a_i\eta_{w_i} σ k = 2 1 i ∑ a i η w i has exactly the same masses as σ \sigma σ k k k
As k → ∞ k\to\infty k → ∞ σ k \sigma_k σ k σ \sigma σ σ \sigma σ β ± 1 \beta^{\pm1} β ± 1
Finally, the return paths are reduced as observed after Proposition 5.6. Fixing their first and last edges is equivalent to fixing the intervening segment, so their full stabilizers in P P P S + S_+ S + S − S_- S − v v v M M M M M M x ↦ v − b x v b x\mapsto v^{-b}xv^b x ↦ v − b x v b x ↦ v b x v − b x\mapsto v^bxv^{-b} x ↦ v b x v − b β \beta β β − 1 \beta^{-1} β − 1 □ \square □
Proposition 5.8 (Zero mass on a maximal carrier). For every maximal carrier M M M
μ ( ∂ 2 Λ M ) = 0. \mu(\partial^2\Lambda M)=0. μ ( ∂ 2 Λ M ) = 0. The same holds for every P P P
Proof. Use the boundary embedding of M M M P P P μ \mu μ ∂ 2 Λ M \partial^2\Lambda M ∂ 2 Λ M ∂ 2 P \partial^2P ∂ 2 P M M M σ \sigma σ μ \mu μ ∂ 2 M \partial^2M ∂ 2 M ∂ 2 P \partial^2P ∂ 2 P μ \mu μ Θ + \Theta_{+} Θ + ∂ 2 Λ M \partial^{2}\Lambda M ∂ 2 Λ M β \beta β β ∗ σ = σ \beta_{*}\sigma=\sigma β ∗ σ = σ (17) gives
λ ∥ σ ∥ M ≤ 2 ∥ σ ∥ M . \lambda\lVert\sigma\rVert_{M}\le2\lVert\sigma\rVert_{M}. λ ∥ σ ∥ M ≤ 2 ∥ σ ∥ M . Since λ > 4 \lambda>4 λ > 4 σ = 0 \sigma=0 σ = 0 μ \mu μ □ \square □
This is a measure conclusion, not a support conclusion: a closed set of measure zero can still contain lines of Ω \Omega Ω
Expansion and periodic lines in the support
We retain the nonzero stationary current μ \mu μ Ω ⊂ ∂ 2 P \Omega\subset\partial^{2}P Ω ⊂ ∂ 2 P U \mathcal{U} U Ω \Omega Ω U \mathcal{U} U Ω \Omega Ω
The obstruction is a possible carrier line in the support despite its zero measure. We will expand a positive-measure cylinder near such a line along a fixed return. If the whole cylinder follows the return forever, its translated images concentrate on the zero-mass carrier set. If a line leaves the prescribed return domain, it follows a competing window; the cyclic intersection of the two window stabilizers forces a periodic middle in their long common segment. Lemma 6.3 records this alternative quantitatively. We first use it to produce periodic support from carrier support, then use whole-period pullback to rule out periodic support itself.
The window results provide the finite data for this argument. For each fixed shape there are finitely many representative cores of noncyclic window stabilizers; only finitely many actual windows have their hulls meeting a fixed ball. Distinct actual windows of that shape have at most cyclic common stabilizer, and an irrational endpoint determines at most one window (Proposition 4.8 and Lemma 4.9). We will also use the fixed small subgraph of the selected cores Γ n \Gamma_{n} Γ n
Let M ≤ P M\le P M ≤ P β \beta β M M M λ > 4 \lambda>4 λ > 4 Θ + \Theta_{+} Θ + Θ − \Theta_{-} Θ −
∂ 2 Λ S + and ∂ 2 Λ S − , M ≤ S + , S − ≤ P . \partial^{2}\Lambda S_{+}\quad\text{and}\quad\partial^{2}\Lambda S_{-},\qquad M\le S_{+},S_{-}\le P. ∂ 2 Λ S + and ∂ 2 Λ S − , M ≤ S + , S − ≤ P . Here and below ∂ 2 Λ K \partial^{2}\Lambda K ∂ 2 Λ K K K K M M M β \beta β β − 1 \beta^{-1} β − 1 M M M
λ ∣ w ∣ M ≤ max { ∣ β ( w ) ∣ M , ∣ β − 1 ( w ) ∣ M } ( w ∈ M ) . (18) \lambda\lvert w\rvert_{M}\le\max\{\lvert\beta(w)\rvert_{M},\lvert\beta^{-1}(w)\rvert_{M}\}\qquad(w\in M). \tag*{(18)} λ ∣ w ∣ M ≤ max {∣ β ( w ) ∣ M , ∣ β − 1 ( w ) ∣ M } ( w ∈ M ) . (18) The same inequality holds for cyclic lengths. Moreover, Proposition 5.8 gives
μ ( ∂ 2 Λ M ) = 0. (19) \mu(\partial^{2}\Lambda M)=0. \tag*{(19)} μ ( ∂ 2 Λ M ) = 0. (19) All transfers preserve μ \mu μ U \mathcal{U} U U \mathcal{U} U
We call a line of the P P P M M M Λ M \Lambda_M Λ M M M M M M M P P P
Write C Q C_Q C Q Q Q Q
Lemma 6.1 (Nearness and order under a fixed transfer). Let a : Q → F a: Q \to F a : Q → F b : Q → F ′ b: Q \to F' b : Q → F ′ D ≥ 0 D \ge0 D ≥ 0 E , T ≥ 0 E,T \ge0 E , T ≥ 0 L L L C Q C_Q C Q L a , L b L_a,L_b L a , L b C F , C F ′ C_F,C_{F'} C F , C F ′ q ∈ Q q \in Q q ∈ Q d ( a ( q ) , L a ) ≤ D d(a(q),L_a) \le D d ( a ( q ) , L a ) ≤ D
d ( b ( q ) , L b ) ≤ E . d(b(q),L_b) \le E. d ( b ( q ) , L b ) ≤ E . If two such elements q 1 , q 2 q_1,q_2 q 1 , q 2 L a L_a L a T T T L b L_b L b L a L_a L a L b L_b L b L , q 1 , q 2 L,q_1,q_2 L , q 1 , q 2
Proof. Fix Hausdorff bounds between each tightened image line and the images of vertices of L L L u ∈ L u \in L u ∈ L a a a L a L_a L a a ( q ) a(q) a ( q ) a a a d Q ( q , u ) d_Q(q,u) d Q ( q , u ) D D D a a a b b b
For the order assertion, first consider actual vertices u < v u < v u < v f f f v v v f ( u ) f(u) f ( u ) f ( z ) f(z) f ( z ) z z z v v v z z z f ( v ) f(v) f ( v ) f ( u ) f(u) f ( u )
c f d Q ( u , v ) − C f c_f d_Q(u,v) - C_f c f d Q ( u , v ) − C f for fixed c f > 0 , C f ≥ 0 c_f > 0,C_f \ge0 c f > 0 , C f ≥ 0
Apply this first to a a a b b b q i q_i q i u i ∈ L u_i \in L u i ∈ L a ( q i ) a(q_i) a ( q i ) u 1 < u 2 u_1 < u_2 u 1 < u 2 b ( q i ) b(q_i) b ( q i ) D D D
We will also use two elementary properties of trees. If x , y x,y x , y D D D L L L [ x , y ] [x,y] [ x , y ] D D D L L L x , y x,y x , y L L L L L L d ( x , y ) ≤ 2 D d(x,y) \le2D d ( x , y ) ≤ 2 D D D D
Lemma 6.2 (Expansion with fixed trimming). Fix a maximal carrier M M M Q ≥ 0 Q \ge0 Q ≥ 0 L ∗ > 0 L_* > 0 L ∗ > 0 L L L M M M x , y x,y x , y L L L d M ( x , y ) = l ≥ L ∗ d_M(x,y)=l \ge L_* d M ( x , y ) = l ≥ L ∗ A ⊂ U \mathcal{A} \subset\mathcal{U} A ⊂ U Q Q Q x , y x,y x , y C P C_P C P ϵ ∈ { + 1 , − 1 } \epsilon\in\{+1,-1\} ϵ ∈ { + 1 , − 1 }
d M ( β ϵ x , β ϵ y ) ≥ λ l . d_M(\beta^\epsilon x,\beta^\epsilon y) \ge\lambda l. d M ( β ϵ x , β ϵ y ) ≥ λ l . If every line of A \mathcal{A} A ∂ 2 Λ S ϵ \partial^2\Lambda S_\epsilon ∂ 2 Λ S ϵ A ′ = Θ ϵ ( A ) \mathcal{A}'=\Theta_\epsilon(\mathcal{A}) A ′ = Θ ϵ ( A ) U \mathcal{U} U x ′ , y ′ x',y' x ′ , y ′ β ϵ L \beta^\epsilon L β ϵ L
d P ( x ′ , ℓ ) , d P ( y ′ , ℓ ) ≤ Q ( ℓ ∈ A ′ ) , d M ( x ′ , y ′ ) ≥ 2 l . d_P(x',\ell),d_P(y',\ell) \le Q \qquad(\ell\in\mathcal{A}'), \qquad d_M(x',y') \ge2l. d P ( x ′ , ℓ ) , d P ( y ′ , ℓ ) ≤ Q ( ℓ ∈ A ′ ) , d M ( x ′ , y ′ ) ≥ 2 l . The same sign satisfies
d M ( β ϵ x ′ , β ϵ y ′ ) ≥ λ d M ( x ′ , y ′ ) . d_M(\beta^\epsilon x',\beta^\epsilon y') \ge\lambda d_M(x',y'). d M ( β ϵ x ′ , β ϵ y ′ ) ≥ λ d M ( x ′ , y ′ ) . The constants are independent of L , x , y , A L,x,y,\mathcal{A} L , x , y , A
Proof. Choose Q Q Q M M M P P P C 1 C_1 C 1 u = β ϵ x u=\beta^\epsilon x u = β ϵ x z = β ϵ y z=\beta^\epsilon y z = β ϵ y C 1 C_1 C 1 A ′ \mathcal{A}' A ′
Choose lower embedding constants a > 0 , b ≥ 0 a>0,b\ge0 a > 0 , b ≥ 0
d P ( g , h ) ≥ a d M ( g , h ) − b ( g , h ∈ M ) . d_P(g,h) \ge a d_M(g,h)-b \qquad(g,h \in M). d P ( g , h ) ≥ a d M ( g , h ) − b ( g , h ∈ M ) . Let B B B β ± 1 \beta^{\pm1} β ± 1 M M M R R R
R > B , a R − b − Q > C 1 . R>B,\qquad aR-b-Q>C_1. R > B , a R − b − Q > C 1 . Delete R R R [ u , z ] [u,z] [ u , z ] C M C_M C M x ′ , y ′ x',y' x ′ , y ′ (6.3) puts them on the tightened line β ϵ L \beta^\epsilon L β ϵ L P P P Q Q Q [ u , z ] P [u,z]_P [ u , z ] P v v v a R − b − Q > C 1 aR-b-Q>C_1 a R − b − Q > C 1 [ u , z ] P [u,z]_P [ u , z ] P A ′ \mathcal{A}' A ′ x ′ , y ′ x',y' x ′ , y ′ Q Q Q
Put
K = max { Lip ( β ) , Lip ( β − 1 ) } . K=\max\{\operatorname{Lip}(\beta),\operatorname{Lip}(\beta^{-1})\}. K = max { Lip ( β ) , Lip ( β − 1 )} . The new length l ′ l' l ′
l ′ ≥ λ l − 2 R , d M ( β − ϵ x ′ , β − ϵ y ′ ) ≤ l + 2 K R . l' \ge\lambda l-2R,\qquad d_M(\beta^{-\epsilon}x',\beta^{-\epsilon}y') \le l+2KR. l ′ ≥ λ l − 2 R , d M ( β − ϵ x ′ , β − ϵ y ′ ) ≤ l + 2 K R . Choose L ∗ L_* L ∗ l ≥ L ∗ l \ge L_* l ≥ L ∗
λ l − 2 R ≥ 2 l , l + 2 K R < 2 λ l . \lambda l-2R \ge2l,\qquad l+2KR<2\lambda l. λ l − 2 R ≥ 2 l , l + 2 K R < 2 λ l . Then the opposite direction at the new pair has length strictly less than λ l ′ \lambda l' λ l ′ (18) forces expansion in the original direction. Measure preservation and membership in U \mathcal{U} U
A failed return produces a periodic segment
Fix the carrier M M M M M M Ω \Omega Ω
Lemma 6.3 (Finite failure of an expanding return). There are constants L 0 ≥ L ∗ L_0 \ge L_\ast L 0 ≥ L ∗ c > 0 c > 0 c > 0 C 2 ≥ 0 C_2 \ge0 C 2 ≥ 0 D ≥ 1 D \ge1 D ≥ 1 η ∈ Ω ∩ ∂ 2 Λ M \eta\in\Omega\cap\partial^2\Lambda_M η ∈ Ω ∩ ∂ 2 Λ M L L L M M M x 0 < y 0 x_0 < y_0 x 0 < y 0 L L L l 0 = d M ( x 0 , y 0 ) ≥ L 0 l_0 = d_M(x_0,y_0) \ge L_0 l 0 = d M ( x 0 , y 0 ) ≥ L 0 ϵ ∈ { + 1 , − 1 } \epsilon\in\{+1,-1\} ϵ ∈ { + 1 , − 1 }
d M ( β ϵ x 0 , β ϵ y 0 ) ≥ λ l 0 . d_M(\beta^\epsilon x_0,\beta^\epsilon y_0) \ge\lambda l_0. d M ( β ϵ x 0 , β ϵ y 0 ) ≥ λ l 0 . Then there are n ≥ 0 n \ge0 n ≥ 0 ℓ ∈ U \ell\in\mathcal{U} ℓ ∈ U P P P ζ \zeta ζ D D D ℓ \ell ℓ ζ \zeta ζ P P P η n \eta_n η n β ϵ n L \beta^{\epsilon n}L β ϵ n L
c 2 n l 0 − C 2 . c2^n l_0-C_2. c 2 n l 0 − C 2 . The endpoints of ζ \zeta ζ Θ ϵ \Theta_\epsilon Θ ϵ S ϵ S_\epsilon S ϵ
Proof. Choose L 0 ≥ L ∗ L_0 \ge L_\ast L 0 ≥ L ∗ x 0 , y 0 x_0,y_0 x 0 , y 0 η \eta η L 0 L_0 L 0 η ∈ Ω \eta\in\Omega η ∈ Ω U \mathcal{U} U A 0 \mathcal{A}_0 A 0 Q Q Q x 0 , y 0 x_0,y_0 x 0 , y 0
Attempt to apply Θ ϵ \Theta_\epsilon Θ ϵ n n n failure if A n \mathcal{A}_n A n ∂ 2 Λ S ϵ \partial^2\Lambda_{S_\epsilon} ∂ 2 Λ S ϵ A n + 1 = Θ ϵ ( A n ) \mathcal{A}_{n+1}=\Theta_\epsilon(\mathcal{A}_n) A n + 1 = Θ ϵ ( A n ) x n , y n x_n,y_n x n , y n β ϵ n L \beta^{\epsilon n}L β ϵ n L l n l_n l n 2 n l 0 2^n l_0 2 n l 0 A n \mathcal{A}_n A n Q Q Q δ = μ ( A 0 ) > 0 \delta=\mu(\mathcal{A}_0)>0 δ = μ ( A 0 ) > 0
The process cannot continue forever. Choose an M M M a n a_n a n [ x n , y n ] M [x_n,y_n]_M [ x n , y n ] M A n \mathcal{A}_n A n a n − 1 a_n^{-1} a n − 1 [ x n , y n ] P [x_n,y_n]_P [ x n , y n ] P A n \mathcal{A}_n A n a n a_n a n n n n K \mathcal{K} K δ \delta δ
Every subsequential limit of these lines belongs to ∂ 2 Λ M \partial^2\Lambda_M ∂ 2 Λ M M M M M M M P P P Λ M \Lambda_M Λ M
Consequently every relative neighborhood of K ∩ ∂ 2 Λ M \mathcal{K}\cap\partial^2\Lambda_M K ∩ ∂ 2 Λ M K \mathcal{K} K K \mathcal{K} K K \mathcal{K} K
μ ( K ∩ ∂ 2 Λ M ) ≥ δ , \mu(\mathcal{K}\cap\partial^2\Lambda_M)\ge\delta, μ ( K ∩ ∂ 2 Λ M ) ≥ δ , contrary to (6.2).
A failure produces a bounded-period axis. Suppose failure occurs at time n n n ℓ ∈ A n ∖ ∂ 2 Λ S ϵ \ell\in\mathcal{A}_n\setminus\partial^2\Lambda_{S_\epsilon} ℓ ∈ A n ∖ ∂ 2 Λ S ϵ K K K ℓ \ell ℓ Λ S ϵ \Lambda S_{\epsilon} Λ S ϵ
Both ℓ \ell ℓ η n \eta_n η n β ϵ n L \beta_{\epsilon}^{n}L β ϵ n L Q Q Q x n , y n x_n,y_n x n , y n x n , y n ∈ M ≤ S ϵ x_n,y_n \in M \leq S_{\epsilon} x n , y n ∈ M ≤ S ϵ Hull ( Λ S ϵ ) \operatorname{Hull}(\Lambda S_{\epsilon}) Hull ( Λ S ϵ ) d P ( 1 , Hull ( Λ S ϵ ) ) d_P(1,\operatorname{Hull}(\Lambda S_{\epsilon})) d P ( 1 , Hull ( Λ S ϵ ))
H ϵ = max { Q , d P ( 1 , Hull ( Λ S ϵ ) ) } . H_{\epsilon}=\max\{Q,d_P(1,\operatorname{Hull}(\Lambda S_{\epsilon}))\}. H ϵ = max { Q , d P ( 1 , Hull ( Λ S ϵ ))} . Deleting H ϵ H_{\epsilon} H ϵ [ x n , y n ] P [x_n,y_n]_P [ x n , y n ] P ℓ \ell ℓ η n \eta_n η n Hull ( Λ K ) \operatorname{Hull}(\Lambda K) Hull ( Λ K ) ℓ \ell ℓ
where a > 0 , C ≥ 0 a>0,C\geq0 a > 0 , C ≥ 0 K K K
Read this segment in the two finite stabilizer cores. Their fiber product pairs core states, with an edge for each pair of equally labeled edges. At an actual vertex z z z z − 1 z^{-1} z − 1 D ≥ 1 D\geq1 D ≥ 1
Increase L 0 L_0 L 0 a L 0 − C > 2 D + 1 aL_0-C>2D+1 a L 0 − C > 2 D + 1 D D D a l n − C − 2 D a l_n-C-2D a l n − C − 2 D ζ \zeta ζ P P P D D D l n ≥ 2 n l 0 l_n\geq2n l_0 l n ≥ 2 n l 0 (6.5) , with constants enlarged uniformly over the two signs. The line ℓ \ell ℓ U U U
The failure need not occur on a set of positive measure. Its chosen line already belongs to U ⊂ Ω U\subset\Omega U ⊂ Ω Figure 2 shows how the two actual windows produce the periodic middle.
Figure 2. At a failed return, η n \eta_n η n ℓ \ell ℓ ζ \zeta ζ ℓ \ell ℓ
Proposition 6.4. If Ω \Omega Ω M M M Ω \Omega Ω
Proof. Translate the given line into ∂ 2 Λ M \partial^2\Lambda M ∂ 2 Λ M M M M (18) . Apply Lemma 6.3, passing to a subsequence with the same sign. Its support line and bounded-period axis share a segment whose length tends to infinity.
Recenter this shared segment by an M M M η n \eta_n η n D D D D D D ℓ ∈ Ω \ell\in\Omega ℓ ∈ Ω Ω \Omega Ω Ω \Omega Ω
Pulling back whole periods
The next lemma controls the number of periods in a common segment. Controlling only its metric length would be insufficient, since a period can grow under an inverse automorphism.
Lemma 6.5 (Bounded loss of periods). Let ψ \psi ψ be an automorphism of a finitely generated free group M M M with a fixed basis. There is an integer C 4 ≥ 0 C_4 \ge0 C 4 ≥ 0 depending only on ψ \psi ψ with the following property. Suppose two lines L 1 , L 2 L_1, L_2 L 1 , L 2 of C M C_M C M share a segment [ u , z k u ] [u,z^k u] [ u , z k u ] where u u u is a vertex, z z z translates L 2 L_2 L 2 and k > C 4 k > C_4 k > C 4 Their tightened ψ \psi ψ images share a segment consisting of k − C 4 k-C_4 k − C 4 whole ψ ( z ) \psi(z) ψ ( z ) periods. If z z z is primitive as a translation in M M M then so is ψ ( z ) \psi(z) ψ ( z )
Proof. Choose an integer B ≥ 0 B \ge0 B ≥ 0 B B B ψ \psi ψ q = 2 B + 1 q = 2B + 1 q = 2 B + 1 C 4 = 2 q C_4 = 2q C 4 = 2 q
Write L i ′ = ψ L i L'_i = \psi L_i L i ′ = ψ L i z ′ = ψ ( z ) z' = \psi(z) z ′ = ψ ( z ) L 2 ′ L'_2 L 2 ′ z ′ z' z ′ a a a ψ ( u ) \psi(u) ψ ( u )
π L 2 ′ ( ψ ( z k u ) ) = π L 2 ′ ( z ′ k ψ ( u ) ) = z ′ k a , \pi_{L'_2}\left(\psi(z^k u)\right)=\pi_{L'_2}\left(z'^k\psi(u)\right)=z'^k a, π L 2 ′ ( ψ ( z k u ) ) = π L 2 ′ ( z ′ k ψ ( u ) ) = z ′ k a , because projection onto an invariant line commutes with its translation. Both endpoint images lie within B B B a , z ′ k a a,z'^k a a , z ′ k a 2 B 2B 2 B L 1 ′ L'_1 L 1 ′
The translation length of z ′ z' z ′ z ′ q a z'^q a z ′ q a z ′ k − q a z'^{k-q}a z ′ k − q a 2 B 2B 2 B [ a , z ′ k a ] [a,z'^k a] [ a , z ′ k a ] L 1 ′ L'_1 L 1 ′ L 2 ′ L'_2 L 2 ′ k − 2 q = k − C 4 k-2q=k-C_4 k − 2 q = k − C 4 z ′ z' z ′
Lemma 6.6 (Agreement of periodic labels). Two bi-infinite label sequences with respective positive integer periods a , b a,b a , b that agree on a + b a+b a + b consecutive positions agree everywhere. Consequently two oriented periodic lines in a labeled Cayley tree with such a common segment are the same actual line.
Proof. Number the common positions 0 , … , a + b − 1 0,\ldots,a+b-1 0 , … , a + b − 1 0 ≤ i < a 0\le i<a 0 ≤ i < a i i i i + b i+b i + b ( i + b ) m o d a (i+b)\bmod a ( i + b ) mod a b b b a a a d = gcd ( a , b ) d=\gcd(a,b) d = g cd( a , b ) b b b d d d □ \square □
Lemma 6.7 (A bounded-period axis near a carrier). Let M ≤ P M\le P M ≤ P D ≥ 1 D\ge1 D ≥ 1 L D , B D L_D,B_D L D , B D P P P ζ \zeta ζ D D D L D L_D L D M M M ζ \zeta ζ M M M M M M M M M B D B_D B D
Proof. Let V V V M M M P P P L D = ( V + 1 ) D L_D=(V+1)D L D = ( V + 1 ) D Hull ( Λ M ) \operatorname{Hull}(\Lambda M) Hull ( Λ M ) r r r P P P ζ \zeta ζ V + 1 V+1 V + 1 r i z 0 r^i z_0 r i z 0
M r i z 0 = M r j z 0 ( 0 ≤ i < j ≤ V ) . Mr^i z_0=Mr^j z_0\qquad(0\le i<j\le V). M r i z 0 = M r j z 0 ( 0 ≤ i < j ≤ V ) . Cancelling z 0 z_0 z 0 r j − i ∈ M r^{j-i}\in M r j − i ∈ M M M M Λ M \Lambda M Λ M
If z z z M M M r r r M M M z = r q z=r^q z = r q 1 ≤ q ≤ V 1\le q\le V 1 ≤ q ≤ V
∣ g ∣ M ≤ A ∣ g ∣ P + B ( g ∈ M ) . |g|_M\le A|g|_P+B\qquad(g\in M). ∣ g ∣ M ≤ A ∣ g ∣ P + B ( g ∈ M ) . Apply this to powers of z z z
ℓ M ( z ) ≤ A ℓ P ( z ) ≤ A V D . \ell_M(z)\le A\ell_P(z)\le AVD. ℓ M ( z ) ≤ A ℓ P ( z ) ≤ A V D . Thus B D = A V D B_D=AVD B D = A V D □ \square □
Exclusion of periodic support lines
Proposition 6.8. The support Ω \Omega Ω
Proof. Suppose η 0 ∈ Ω \eta_0\in\Omega η 0 ∈ Ω
Step 1: construct a trajectory and locate a maximal carrier. We first construct one compatible trajectory for η 0 \eta_0 η 0
Approximate η 0 \eta_0 η 0 U \mathcal{U} U h s h_s h s h 0 = 1 h_0=1 h 0 = 1 η s ∈ Ω \eta_s\in\Omega η s ∈ Ω P P P r s r_s r s η s \eta_s η s h s + 1 = h s p s q s h_{s+1}=h_sp_sq_s h s + 1 = h s p s q s p s , q s ∈ P p_s,q_s\in P p s , q s ∈ P m s m_s m s r s m s ∈ p s I p s − 1 r_s^{m_s}\in p_sIp_s^{-1} r s m s ∈ p s I p s − 1 r s + 1 n s r_{s+1}^{n_s} r s + 1 n s n s n_s n s r s + 1 n s ∈ q s − 1 J q s r_{s+1}^{n_s}\in q_s^{-1}Jq_s r s + 1 n s ∈ q s − 1 J q s
Recall the finite ambiguity dictionary of Lemma 4.2: if either minimum exponent exceeds one, or if one of these axes belongs to two distinct domain cosets on its side, one of its primitive root classes lies in a fixed finite set, up to conjugacy and inversion. Infinitely many such indices would repeat a root class along this actual reduced path. Lemma 4.3 would give a tree-hyperbolic element commensurating a nontrivial elliptic cyclic group, contrary to BS-freeness. Therefore only finitely many indices in the entire bilateral trajectory are exceptional in either way.
It follows that one can choose positive integers k s k_s k s s ∈ Z s\in\mathbb{Z} s ∈ Z
m s ∣ k s , k s + 1 = n s k s / m s . m_s\mid k_s,\qquad k_{s+1}=n_sk_s/m_s. m s ∣ k s , k s + 1 = n s k s / m s . Only finitely many ratios differ from one. Choose k 0 k_0 k 0
h s r s k s h s − 1 h_sr_s^{k_s}h_s^{-1} h s r s k s h s − 1 is one fixed nontrivial element, independent of s s s
Fix s s s r s k s r_s^{k_s} r s k s Γ n \Gamma_n Γ n h s e h_se h s e E j ′ = h s − 1 E s + j E'_j=h_s^{-1}E_{s+j} E j ′ = h s − 1 E s + j E 0 ′ = e E'_0=e E 0 ′ = e Γ n \Gamma_n Γ n η s \eta_s η s P P P M M M ∂ 2 Λ M \partial^2\Lambda_M ∂ 2 Λ M
Step 2: follow expanding return iterates. The subsequent forward path agrees with the path fixed by M M M β \beta β Ω \Omega Ω
Let a j ≥ 1 a_j\ge1 a j ≥ 1 M M M j j j β \beta β a j + 1 ≥ λ a j a_{j+1}\ge\lambda a_j a j + 1 ≥ λ a j j ≥ 1 j\ge1 j ≥ 1 a j − 1 ≥ λ a j a_{j-1}\ge\lambda a_j a j − 1 ≥ λ a j d 0 d_0 d 0
Let z z z M M M x 0 x_0 x 0 y 0 = z k x 0 y_0=z^kx_0 y 0 = z k x 0 k k k l 0 = k d 0 l_0=kd_0 l 0 = k d 0 ℓ M ( β z ) ≥ λ d 0 \ell_M(\beta z)\ge\lambda d_0 ℓ M ( β z ) ≥ λ d 0
d M ( β x 0 , β y 0 ) = d M ( β x 0 , ( β z ) k β x 0 ) ≥ k ℓ M ( β z ) ≥ λ l 0 . d_M(\beta x_0,\beta y_0)=d_M(\beta x_0,(\beta z)^k\beta x_0)\ge k\ell_M(\beta z)\ge\lambda l_0. d M ( β x 0 , β y 0 ) = d M ( β x 0 , ( β z ) k β x 0 ) ≥ k ℓ M ( β z ) ≥ λ l 0 . This verifies the based-segment hypothesis even if β x 0 \beta x_0 β x 0 β z \beta z β z l 0 ≥ L 0 l_0 \ge L_0 l 0 ≥ L 0 n n n (6.5) with the n n n
Step 3: pull a failure axis back to the initial period. The resulting periodic P P P ζ \zeta ζ M M M (6.5) . Take l 0 l_0 l 0 (6.5) exceeds the fixed threshold in Lemma 6.7, already at n = 0 n = 0 n = 0 ζ \zeta ζ M M M M M M (6.7) .
The two corresponding M M M c 0 2 n l 0 − C 0 c_0 2^n l_0 - C_0 c 0 2 n l 0 − C 0 c 0 > 0 c_0 > 0 c 0 > 0 C 0 ≥ 0 C_0 \ge0 C 0 ≥ 0 P P P M M M P P P M M M M M M P P P
The uniformly bounded primitive period of ζ \zeta ζ
c ′ 2 n l 0 − C 3 c'2^n l_0-C_3 c ′ 2 n l 0 − C 3 whole primitive ζ \zeta ζ c ′ > 0 c' > 0 c ′ > 0 C 3 ≥ 0 C_3 \ge0 C 3 ≥ 0 ψ = β − 1 \psi= \beta^{-1} ψ = β − 1 n n n
c ′ 2 n l 0 − C 3 − n C 4 c'2^n l_0-C_3-nC_4 c ′ 2 n l 0 − C 3 − n C 4 whole periods remain. Choose the initial length so large that
c ′ l 0 > max { C 4 , C 3 + d 0 + 1 } . c'l_0>\max\{C_4,\ C_3+d_0+1\}. c ′ l 0 > max { C 4 , C 3 + d 0 + 1 } . Since 2 n ≥ n + 1 2^n \ge n+1 2 n ≥ n + 1 (6.10) exceeds d 0 + 1 d_0+1 d 0 + 1 n n n
The primitive period of the second line after these pullbacks may be large; call it b ≥ 1 b \ge1 b ≥ 1 d 0 + 1 d_0+1 d 0 + 1 ( d 0 + 2 ) b ≥ d 0 + b (d_0+2)b \ge d_0+b ( d 0 + 2 ) b ≥ d 0 + b ζ \zeta ζ M M M β n \beta^n β n M M M P P P
The trajectory construction on the entire support
We have excluded periodic support points without assuming that all support points admit unique transfers. We can now prove precisely that extension, together with the uniform bound needed to count special rays.
Corollary 6.9. The following assertions hold for the closed support Ω \Omega Ω
No line of Ω \Omega Ω B 0 B_0 B 0 Ω \Omega Ω Γ n \Gamma_n Γ n B 0 B_0 B 0
Every endpoint of every line of Ω \Omega Ω
Every line of Ω \Omega Ω Γ n \Gamma_n Γ n Ω \Omega Ω
Proof. An immersed bi-infinite path in a finite cyclic small component is periodic, and such a line is excluded by Proposition 6.8. A line carried by a noncyclic small component is carried, up to conjugacy, by a maximal noncyclic carrier. Proposition 6.4 would then produce a periodic support line, also impossible. A tree component cannot carry an immersed bi-infinite path. This proves the first assertion about whole lines.
If small-only segments of lifted support lines had unbounded length, center progressively longer such segments at an actual vertex and translate that vertex to the identity in P P P Ω \Omega Ω B 0 B_0 B 0
A rational endpoint is represented by an eventually periodic ray. If such a ray lay on a support line, translating farther and farther along its periodic tail would give a periodic line as a limit of support lines. Closedness of Ω \Omega Ω
The existence of lifts to each finite core was already proved in Lemma 5.1: it follows from conull lifts, density, and closedness of liftability into a fixed finite graph. Each such lift gives an actual window path carrying the line’s endpoints. The irrational-endpoint uniqueness in Proposition 4.8 makes these paths compatible when a window is shortened. They therefore give a unique bilateral trajectory.
It remains to check the support of transferred lines. Approximate a given support line by lines of U \mathcal{U} U U ⊂ Ω \mathcal{U} \subset\Omega U ⊂ Ω Ω \Omega Ω
Special rays and the end of the combination proof
We finish the proof of Theorem 3.1. Suppose that G G G μ \mu μ Ω \Omega Ω U \mathcal{U} U Ω \Omega Ω Ω \Omega Ω Ω \Omega Ω Γ n \Gamma_n Γ n Ω \Omega Ω
The remaining argument detects branching in the language of Ω \Omega Ω
Finitely many rays with two pasts
Fix the basis of P P P T P T_P T P occurring if it labels a positive ray in a line of Ω \Omega Ω ρ \rho ρ left-special if there are distinct basis letters a , b a,b a , b a ρ a\rho a ρ b ρ b\rho b ρ 1 1 1 Ω \Omega Ω
Lemma 7.1. The set of left-special occurring words is finite and nonempty.
Proof. Let ρ \rho ρ Γ n \Gamma_n Γ n
Let B B B ρ \rho ρ B B B B B B Γ n \Gamma_n Γ n ρ \rho ρ
Suppose now that no occurring ray is left-special. Then there is an integer N N N N N N Ω \Omega Ω
Form the finite directed graph whose vertices are occurring words of length N N N N + 1 N+1 N + 1 N N N N N N Ω \Omega Ω □ \square □
A return map with positive lag
Let E \mathcal{E} E P P P E \mathcal{E} E P P P P P P E \mathcal{E} E
Choose a left-special word based at 1 1 1 ξ \xi ξ E \mathcal{E} E p > 0 p>0 p > 0 p p p P P P v ∈ G v\in G v ∈ G p p p
Θ : S ⟶ T , Θ ( x ) = v − 1 x v , S = P ∩ v P v − 1 , T = P ∩ v − 1 P v , \Theta:S\longrightarrow T,\qquad\Theta(x)=v^{-1}xv,\qquad S=P\cap vPv^{-1},\qquad T=P\cap v^{-1}Pv, Θ : S ⟶ T , Θ ( x ) = v − 1 xv , S = P ∩ v P v − 1 , T = P ∩ v − 1 P v , whose boundary map fixes ξ \xi ξ S S S T T T P P P μ \mu μ U \mathcal{U} U
Write r 0 = 1 , r 1 , r 2 , … r_0=1,r_1,r_2,\ldots r 0 = 1 , r 1 , r 2 , … ξ \xi ξ T P T_P T P ξ ∈ Λ S \xi\in\Lambda S ξ ∈ Λ S S S S s n ∈ S s_n\in S s n ∈ S K ≥ 0 K\geq0 K ≥ 0 d ( s n , r n ) ≤ K d(s_n,r_n)\leq K d ( s n , r n ) ≤ K n n n Θ \Theta Θ T T T ξ \xi ξ f ( n ) f(n) f ( n )
d ( Θ ( s n ) , r f ( n ) ) ≤ C , f ( n ) ⟶ ∞ , ∣ f ( n + 1 ) − f ( n ) ∣ ≤ C . (20) d(\Theta(s_n),r_{f(n)})\leq C,\qquad f(n)\longrightarrow\infty,\qquad|f(n+1)-f(n)|\leq C. \tag*{(20)} d ( Θ ( s n ) , r f ( n ) ) ≤ C , f ( n ) ⟶ ∞ , ∣ f ( n + 1 ) − f ( n ) ∣ ≤ C . (20) after increasing a fixed constant C C C
Lemma 7.2. After replacing Θ \Theta Θ
f ( n ) − n ⟶ + ∞ . f(n)-n\longrightarrow+\infty. f ( n ) − n ⟶ + ∞. Proof. If ∣ f ( n ) − n ∣ |f(n)-n| ∣ f ( n ) − n ∣ s n − 1 Θ ( s n ) s_n^{-1}\Theta(s_n) s n − 1 Θ ( s n ) s n s_n s n s n , s m s_n,s_m s n , s m d ∈ P d\in P d ∈ P Θ ( s n ) = s n d \Theta(s_n)=s_nd Θ ( s n ) = s n d Θ ( s m ) = s m d \Theta(s_m)=s_md Θ ( s m ) = s m d
Θ ( s n s m − 1 ) = ( s n d ) ( s m d ) − 1 = s n s m − 1 . \Theta(s_ns_m^{-1})=(s_nd)(s_md)^{-1}=s_ns_m^{-1}. Θ ( s n s m − 1 ) = ( s n d ) ( s m d ) − 1 = s n s m − 1 . The nontrivial element s n s m − 1 s_ns_m^{-1} s n s m − 1 v v v p ≠ 0 p\neq0 p = 0 Z 2 \mathbb{Z}^2 Z 2
Thus ∣ f ( n ) − n ∣ ⟶ ∞ |f(n)-n|\longrightarrow\infty ∣ f ( n ) − n ∣ ⟶ ∞ f ( n ) − n f(n)-n f ( n ) − n t m ∈ T t_m\in T t m ∈ T r m r_m r m g ( m ) g(m) g ( m ) Θ − 1 ( t m ) \Theta^{-1}(t_m) Θ − 1 ( t m ) r g ( m ) r_{g(m)} r g ( m ) f f f n ( m ) n(m) n ( m )
∣ f ( n ( m ) ) − m ∣ ≤ C , n ( m ) ⟶ ∞ . |f(n(m))-m|\leq C,\qquad n(m)\longrightarrow\infty. ∣ f ( n ( m )) − m ∣ ≤ C , n ( m ) ⟶ ∞. Then t m t_m t m Θ ( s n ( m ) ) \Theta(s_{n(m)}) Θ ( s n ( m ) )
g ( m ) = n ( m ) + O ( 1 ) , g ( m ) − m = n ( m ) − f ( n ( m ) ) + O ( 1 ) ⟶ + ∞ . g(m)=n(m)+O(1),\qquad g(m)-m=n(m)-f(n(m))+O(1)\longrightarrow+\infty. g ( m ) = n ( m ) + O ( 1 ) , g ( m ) − m = n ( m ) − f ( n ( m )) + O ( 1 ) ⟶ + ∞. This proves the assertion for the inverse map.
A cylinder that cannot lose mass
After the possible inversion, use Θ \Theta Θ S S S T T T f f f K K K C C C ( r n ) (r_n) ( r n ) ξ \xi ξ n 0 n_0 n 0 r n 0 r_{n_0} r n 0
Fix s > n 0 s>n_0 s > n 0 [ r s , ξ ) [r_s,\xi) [ r s , ξ ) r n 0 r_{n_0} r n 0 Θ \Theta Θ ξ \xi ξ
Lemma 7.3. For each fixed s > n 0 s>n_0 s > n 0 N s N_s N s N ≥ N s N\geq N_s N ≥ N s Ω \Omega Ω [ r s , r N ] [r_s,r_N] [ r s , r N ] r n 0 r_{n_0} r n 0 Θ \Theta Θ
Proof. For the first assertion, a contrary sequence has a convergent subsequence in the compact space of directed lines through r s r_s r s Ω \Omega Ω [ r s , ξ ) [r_s,\xi) [ r s , ξ ) [ r n 0 , r s ] [r_{n_0},r_s] [ r n 0 , r s ]
For the domain assertion, each competing line has an actual path for the fixed window. Its stabilizer hull contains that line, hence meets the vertex r s r_s r s ξ \xi ξ Θ \Theta Θ
For N ≥ N s N \geq N_s N ≥ N s
U N = { ℓ ∈ U : ℓ contains the directed segment [ r s , r N ] } . U_N = \{\ell\in\mathcal{U} : \ell\text{ contains the directed segment } [r_s,r_N]\}. U N = { ℓ ∈ U : ℓ contains the directed segment [ r s , r N ]} . This set has finite positive μ \mu μ U \mathcal{U} U r n 0 r_{n_0} r n 0 Θ \Theta Θ s n 0 s_{n_0} s n 0 s N s_N s N K K K S ↪ P S \hookrightarrow P S ↪ P Θ : S → P \Theta:S \to P Θ : S → P E E E E E E Θ ( U N ) \Theta(U_N) Θ ( U N ) (20) then puts r f ( n 0 ) r_{f(n_0)} r f ( n 0 ) r f ( N ) r_{f(N)} r f ( N ) E + C E+C E + C N − n 0 − 2 K N-n_0-2K N − n 0 − 2 K N N N
Choose a fixed integer D > E + C D>E+C D > E + C Θ ( U N ) \Theta(U_N) Θ ( U N ) [ r f ( n 0 ) + D , r f ( N ) − D ] [r_{f(n_0)+D},r_{f(N)-D}] [ r f ( n 0 ) + D , r f ( N ) − D ] N N N E E E D D D s s s N N N
Now choose s > max { n 0 , f ( n 0 ) + D } s>\max\{n_0,f(n_0)+D\} s > max { n 0 , f ( n 0 ) + D } N N N N s N_s N s f ( N ) − D ≥ N + 1 f(N)-D\geq N+1 f ( N ) − D ≥ N + 1 N N N U \mathcal{U} U
Θ ( U N ) ⊆ U N + 1 ⊆ U N . \Theta(U_N) \subseteq U_{N+1} \subseteq U_N. Θ ( U N ) ⊆ U N + 1 ⊆ U N . Because Θ \Theta Θ
μ ( U N ) = μ ( Θ ( U N ) ) ≤ μ ( U N + 1 ) ≤ μ ( U N ) . \mu(U_N)=\mu(\Theta(U_N))\leq\mu(U_{N+1})\leq\mu(U_N). μ ( U N ) = μ ( Θ ( U N )) ≤ μ ( U N + 1 ) ≤ μ ( U N ) . All sufficiently late U N U_N U N U ∞ U_\infty U ∞ U ∞ U_\infty U ∞ [ r s , ξ ) [r_s,\xi) [ r s , ξ )
For each n ≥ s n\geq s n ≥ s U ∞ U_\infty U ∞ r n − 1 r_n^{-1} r n − 1 1 1 1 r n − 1 ξ r_n^{-1}\xi r n − 1 ξ n ≠ m n\ne m n = m P P P ξ \xi ξ 1 1 1
We have contradicted the existence of the nonzero stationary current forced by nonhyperbolicity. Therefore G G G
The Magnus induction
We now apply Theorem 3.1 to one-relator groups. The geometric form of the Magnus construction gives an HNN extension whose vertex group has smaller complexity and whose edge groups are simultaneous Magnus subgroups. The induction will establish hyperbolicity. At each stage, Linton’s theorem on Magnus subgroups of hyperbolic one-relator groups supplies the quasiconvexity needed by the combination theorem.
Graph presentations and Magnus subgroups
Let Γ \Gamma Γ λ : S 1 → Γ \lambda:S^1 \to\Gamma λ : S 1 → Γ
X = Γ ∪ λ D 2 , S λ = λ ( S 1 ) . X = \Gamma\cup_{\lambda} D^2,\qquad S_{\lambda} = \lambda(S^1). X = Γ ∪ λ D 2 , S λ = λ ( S 1 ) . We call a connected subgraph Δ ⊂ Γ \Delta\subset\Gamma Δ ⊂ Γ Magnus subgroup if S λ ⊄ Δ S_{\lambda} \not\subset\Delta S λ ⊂ Δ Δ \Delta Δ λ \lambda λ
Lemma 8.1. Let X = Γ ∪ λ D 2 X = \Gamma\cup_{\lambda} D^2 X = Γ ∪ λ D 2 Δ ⊂ Γ \Delta\subset\Gamma Δ ⊂ Γ Magnus subgraph . There is a free basis Σ \Sigma Σ π 1 Γ \pi_1\Gamma π 1 Γ π 1 Δ \pi_1\Delta π 1 Δ Σ \Sigma Σ π 1 Δ → π 1 X \pi_1\Delta\to\pi_1X π 1 Δ → π 1 X
Proof. Choose a spanning tree T Δ T_{\Delta} T Δ Δ \Delta Δ T T T Γ \Gamma Γ T T T Σ \Sigma Σ Δ ∖ T Δ \Delta\setminus T_{\Delta} Δ ∖ T Δ T T T λ \lambda λ λ \lambda λ
Suppose that this word used only the letters from Δ \Delta Δ λ \lambda λ Δ \Delta Δ T Δ T_{\Delta} T Δ λ \lambda λ Δ \Delta Δ [18 ] . □
We use Linton’s theorem in the following form: every Magnus subgroup of a hyperbolic one-relator group is quasiconvex [13 ] . Together with Lemma 8.1, it applies to connected Magnus subgraphs of any graph presentation whose group has already been shown to be hyperbolic.
A finite domain in a cyclic cover
We use the geometric Magnus construction developed in [12 ] , giving the domain and complexity arguments in the form needed here. The construction finds a finite part of a cyclic cover whose adjacent overlaps will supply the two Magnus subgroups in an HNN splitting.
For a graph of rank at least two, choose an epimorphism
ϕ : π 1 Γ ⟶ Z with ϕ ( [ λ ] ) = 0. \phi:\pi_1\Gamma\longrightarrow\mathbb{Z}\quad\text{with}\quad\phi([\lambda])=0. ϕ : π 1 Γ ⟶ Z with ϕ ([ λ ]) = 0. Such a map exists: the abelianized relator imposes at most one equation on the integral characters of a free abelian group of rank at least two, and dividing a nonzero solution by the gcd of its coefficients makes it surjective. Let p : Γ ~ → Γ p:\widetilde{\Gamma}\to\Gamma p : Γ → Γ τ \tau τ domain will mean a finite connected subgraph D ⊂ Γ ~ D\subset\widetilde{\Gamma} D ⊂ Γ c ~ \widetilde{c} c c c c
{ n ∈ Z : τ n c ~ ⊂ D } \{n \in\mathbb{Z} : \tau^n\widetilde{c} \subset D\} { n ∈ Z : τ n c ⊂ D } is a nonempty finite interval. In particular, the translates of a domain cover Γ ~ \widetilde{\Gamma} Γ [12 ] , Lemma 4.7.
Lemma 8.2. Every connected cyclic cover of a finite connected graph has a domain D D D D ∩ τ D D \cap\tau D D ∩ τ D
Proof. Choose a spanning tree downstairs. We may then index the lifted vertices by ( v , n ) (v,n) ( v , n ) n n n d e d_e d e Z \mathbb{Z} Z D N D_N D N [ − N , N ] [-N,N] [ − N , N ] N N N
We verify connectivity, including near the boundary of the interval. Choose a word in the signed increments whose sum is 1 1 1 L L L [ − N , N ] [-N,N] [ − N , N ] a a a a a a N > L + a N>L+a N > L + a D N D_N D N [ − N + 1 , N ] [-N+1,N] [ − N + 1 , N ] D N ∩ τ D N D_N \cap\tau D_N D N ∩ τ D N
Among domains with connected nonempty adjacent overlap, choose D D D D ∩ τ D D \cap\tau D D ∩ τ D D D D D ∩ τ D D \cap\tau D D ∩ τ D D D D e e e τ e ∉ D \tau e \notin D τ e ∈ / D e ∈ D ∩ τ D e \in D \cap\tau D e ∈ D ∩ τ D τ − 1 e \tau^{-1}e τ − 1 e D D D e e e
( D ∖ e ) ∩ τ ( D ∖ e ) = ( D ∩ τ D ) ∖ e , (D \setminus e) \cap\tau(D \setminus e) = (D \cap\tau D) \setminus e, ( D ∖ e ) ∩ τ ( D ∖ e ) = ( D ∩ τ D ) ∖ e , because τ e ∉ D \tau e \notin D τ e ∈ / D
We next enlarge or modify such a domain to contain one lifted relator. Its two overlaps will become the edge groups of the splitting. The construction also accounts for strict decrease of the complexity
c ( Γ , λ ) = ( ∣ λ ∣ − ∣ V ( S ( λ ) ) ∣ , rank π 1 Γ ) , (21) c(\Gamma,\lambda) = (|\lambda|-|V(\mathcal{S}(\lambda))|,\operatorname{rank}\pi_1\Gamma), \tag*{(21)} c ( Γ , λ ) = ( ∣ λ ∣ − ∣ V ( S ( λ )) ∣ , rank π 1 Γ ) , (21) ordered lexicographically. Here ∣ λ ∣ |\lambda| ∣ λ ∣ [12 ] , Section 4.3.
Lemma 8.3. Suppose rank π 1 Γ ≥ 2 \operatorname{rank}\pi_1\Gamma\ge2 rank π 1 Γ ≥ 2 λ \lambda λ π 1 Γ \pi_1\Gamma π 1 Γ U ⊂ Γ ~ U \subset\widetilde{\Gamma} U ⊂ Γ λ ~ : S 1 → U \widetilde{\lambda}\colon S^1 \to U λ : S 1 → U
U − = U ∩ τ − 1 U , U + = U ∩ τ U , Z = U ∪ λ ~ D 2 . U_- = U \cap\tau^{-1}U,\qquad U_+ = U \cap\tau U,\qquad Z = U \cup_{\widetilde{\lambda}} D^2. U − = U ∩ τ − 1 U , U + = U ∩ τ U , Z = U ∪ λ D 2 . Both U − U_{-} U − U + U_{+} U + Z Z Z U − ∩ U + U_{-} \cap U_{+} U − ∩ U + H = π 1 Z H = \pi_{1}Z H = π 1 Z
π 1 X ≅ ⟨ H , t ∣ t A t − 1 = B ⟩ , A = π 1 U − , B = π 1 U + , (22) \pi_{1}X \cong\langle H,t \mid tAt^{-1} = B\rangle,\qquad A = \pi_{1}U_{-},\quad B = \pi_{1}U_{+}, \tag*{(22)} π 1 X ≅ ⟨ H , t ∣ t A t − 1 = B ⟩ , A = π 1 U − , B = π 1 U + , (22) with edge isomorphism induced by τ \tau τ
c ( U , λ ~ ) < c ( Γ , λ ) . c(U,\widetilde{\lambda}) < c(\Gamma,\lambda). c ( U , λ ) < c ( Γ , λ ) . Proof. We first construct U U U [12 ] , Propositions 4.9 and 4.11.
Case 1: the lifted support does not project injectively on vertices. Take D D D
U = ⋃ i = 0 m τ i D . U = \bigcup_{i=0}^{m} \tau^{i}D. U = i = 0 ⋃ m τ i D . where m ≥ 0 m \ge0 m ≥ 0 λ ~ \widetilde{\lambda} λ D D D U U U [ a , b ] [a,b] [ a , b ] D D D [ a , b + m ] [a,b+m] [ a , b + m ] U U U Figure 3 shows how taking adjacent overlaps trims the endpoints of this interval.
Figure 3. Occurrence intervals for one fixed base cell in Case 1, illustrated with b − a = m = 3 b-a=m=3 b − a = m = 3 D D D m = 0 m=0 m = 0 U = D U=D U = D
If m = 0 m=0 m = 0 m ≥ 1 m \ge1 m ≥ 1
U − = ⋃ i = 0 m − 1 τ i D , U + = ⋃ i = 1 m τ i D . U_{-} = \bigcup_{i=0}^{m-1}\tau^{i}D,\qquad U_{+} = \bigcup_{i=1}^{m}\tau^{i}D. U − = i = 0 ⋃ m − 1 τ i D , U + = i = 1 ⋃ m τ i D . Minimality of m m m λ \lambda λ
U − ∩ U + = { D ∩ τ D , m = 1 , ⋃ i = 1 m − 1 τ i D , m ≥ 2. U_{-} \cap U_{+} =
\begin{cases}
D \cap\tau D, & m=1,\\
\bigcup_{i=1}^{m-1}\tau^{i}D, & m \ge2.
\end{cases} U − ∩ U + = { D ∩ τ D , ⋃ i = 1 m − 1 τ i D , m = 1 , m ≥ 2. It is connected and nonempty. In the case m = 0 m = 0 m = 0 λ \lambda λ
Case 2: the lifted support projects injectively on vertices. It then projects isomorphically onto S λ S_\lambda S λ π 1 S λ \pi_1 S_\lambda π 1 S λ T S T_S T S S λ S_\lambda S λ Γ \Gamma Γ
Contract the support downstairs and every support copy upstairs. This gives the connected cyclic cover of the quotient graph Γ / S λ \Gamma/S_\lambda Γ/ S λ T S T_S T S S λ S_\lambda S λ U U U U ∩ τ U U \cap\tau U U ∩ τ U λ \lambda λ
Here the first complexity coordinate is unchanged and the second decreases. To see this directly, for each base open cell its number of occurrences in U U U U ∩ τ U U \cap\tau U U ∩ τ U
χ ( U ) − χ ( U ∩ τ U ) = χ ( Γ ) . \chi(U) - \chi(U \cap\tau U) = \chi(\Gamma). χ ( U ) − χ ( U ∩ τ U ) = χ ( Γ ) . Since the overlap is a tree, χ ( U ) = χ ( Γ ) + 1 \chi(U) = \chi(\Gamma) + 1 χ ( U ) = χ ( Γ ) + 1
rank π 1 U = rank π 1 Γ − 1. \operatorname{rank} \pi_1 U = \operatorname{rank} \pi_1 \Gamma- 1. rank π 1 U = rank π 1 Γ − 1. The HNN splitting. Both constructions now have the required overlaps and complexity decrease. We recover X X X Z Z Z [12 ] . Since ϕ ( [ λ ] ) = 0 \phi([\lambda]) = 0 ϕ ([ λ ]) = 0 Y → X Y \to X Y → X τ i Z \tau^i Z τ i Z Y Y Y Z Z Z W W W W → Y W \to Y W → Y W → Y W \to Y W → Y
Taking the quotient by translation gives a graph of spaces with one vertex space Z Z Z W / ⟨ τ ⟩ → X W/\langle\tau\rangle\to X W / ⟨ τ ⟩ → X Z Z Z H H H π 1 X \pi_1 X π 1 X
Simultaneous Magnus bases and the edge isomorphism. It remains to choose the two Magnus subsets simultaneously and specify their basepoint paths. If the overlap groups are trivial, this is immediate. Otherwise put U 0 = U − ∩ U + U_0 = U_- \cap U_+ U 0 = U − ∩ U + U 0 U_0 U 0 U − U_- U − U + U_+ U + U 0 U_0 U 0 U U U U 0 U_0 U 0 π 1 U \pi_1 U π 1 U C = π 1 U 0 C = \pi_1 U_0 C = π 1 U 0
To specify the stable letter in this nontrivial-edge case, choose x ∈ U 0 x \in U_0 x ∈ U 0 q ⊂ U + q \subset U_+ q ⊂ U + x x x τ x \tau x τ x U + U_+ U + x ∈ U − x \in U_- x ∈ U − t = [ p ( q ) ] t = [p(q)] t = [ p ( q )] a a a U − U_- U − q ( τ a ) q − 1 q(\tau a)q^{-1} q ( τ a ) q − 1 U + U_+ U + t a t − 1 tat^{-1} t a t − 1 x ∈ U − x \in U_- x ∈ U − q ⊂ U q \subset U q ⊂ U x x x τ x \tau x τ x t = [ p ( q ) ] t = [p(q)] t = [ p ( q )] q q q Z Z Z τ Z \tau Z τ Z τ x \tau x τ x p ( q ) p(q) p ( q )
Finally, π 1 U \pi_1 U π 1 U π 1 Γ \pi_1\Gamma π 1 Γ π 1 U \pi_1 U π 1 U π 1 Γ \pi_1\Gamma π 1 Γ
Hyperbolicity and the final reductions
The preceding construction supplies the geometric input to the combination theorem. We now check its algebraic hypotheses.
Proposition 8.4. Let Γ \Gamma Γ λ : S 1 → Γ \lambda:S^1 \to\Gamma λ : S 1 → Γ π 1 Γ \pi_1\Gamma π 1 Γ π 1 ( Γ ∪ λ D 2 ) \pi_1(\Gamma\cup_\lambda D^2) π 1 ( Γ ∪ λ D 2 ) B S BS B S
Proof. We induct on the nonnegative lexicographic complexity (8.1). In graph rank one, the non-power relator generates the infinite cyclic fundamental group, and the quotient is trivial. Rank zero cannot support the stipulated circuit.
In higher rank, apply Lemma 8.3 to obtain
G = ⟨ H , t ∣ t A t − 1 = B ⟩ . G = \langle H,t \mid tAt^{-1} = B\rangle. G = ⟨ H , t ∣ t A t − 1 = B ⟩ . The vertex group H H H G G G B S BS B S [18 ] , Theorem 2. The side groups A A A B B B H H H [13 ] , Theorem 4.7, they are quasiconvex in H H H
Use the simultaneous bases from Lemma 8.3. Collins’s conjugate-intersection theorem [4 ] , Theorem 2, which permits equal Magnus subgroups, says that for M , N ∈ { A , B } M,N \in\{A,B\} M , N ∈ { A , B } g ∈ H g \in H g ∈ H
M ∩ g N g − 1 noncyclic ⟹ g ∈ M N . M \cap gNg^{-1}\ \text{noncyclic}\quad\Longrightarrow\quad g \in MN. M ∩ g N g − 1 noncyclic ⟹ g ∈ M N . This is the first overlap hypothesis of Theorem 3.1. Collins’s direct-intersection theorem [4 ] , Theorem 1 gives
I = A ∩ B = C or I = C ∗ ⟨ u ⟩ , I = A \cap B = C\quad\text{or}\quad I = C * \langle u\rangle, I = A ∩ B = C or I = C ∗ ⟨ u ⟩ , where C C C I I I J = t − 1 I t J = t^{-1}It J = t − 1 I t U − , U + U_-,U_+ U − , U + [12 ] , Lemma 2.10 states that the intersection of the fundamental groups of two such Magnus subgraphs is strongly inert in either side group. It follows that I I I P = A P = A P = A B B B J J J P P P
All hypotheses of Theorem 3.1 have now been verified. Its conclusion makes G G G
Proof of Theorem 1.1. Freely and cyclically reduce the defining relator; conjugating or inverting it does not change its normal closure. If it becomes trivial, the group is a finitely generated free group and is hyperbolic.
If the cyclically reduced relator is a proper power, write it as w k w^k w k w w w k ≥ 2 k \ge2 k ≥ 2 [20 ] , in the form [18 ] , Theorem 4, says that every nonempty freely reduced null word contains a segment of a cyclic conjugate of w ± k w^{\pm k} w ± k ( k − 1 ) ∣ w ∣ (k-1)|w| ( k − 1 ) ∣ w ∣ k = 2 k = 2 k = 2 [2 ] , Theorem III.Γ.2.6.
In the remaining case, realize the cyclically reduced non-power relator as an immersed circuit in the basis rose. Proposition 8.4 applies and proves hyperbolicity under the stated exclusion of B S ( m , n ) BS(m,n) B S ( m , n ) m , n m,n m , n
Proof of Theorem 1.2. Under its non-power graph-presentation hypotheses, Proposition 8.4 makes the ambient group hyperbolic. For any connected subgraph omitting a traversed edge, Lemma 8.1 identifies its fundamental group with an injective Magnus subgroup in a one-relator presentation. Linton’s theorem [13 ] , Theorem 4.7, makes that subgroup quasiconvex. Basepoint paths only conjugate it, so the assertion holds for every choice of such paths.
Virtual free-by-cyclic consequence
We now combine the hyperbolicity theorem with the companion virtual compact specialness theorem and the results of Kielak–Linton.
Corollary 9.1. 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 ⟩⟩ B S ( m , n ) BS(m,n) B S ( m , n ) m , n m,n m , n H ≤ G H \leq G H ≤ G d ≥ 0 d \geq0 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 . Consequently the same 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 \leq\mathbb{Z}. 1 ⟶ K ⟶ H ⟶ Q ⟶ 1 , K free , Q ≤ Z . Thus G G G K K K Q Q Q Proof. If G G G H = 1 H = 1 H = 1 F d = F 0 = 1 F_d = F_0 = 1 F d = F 0 = 1 Z \mathbb{Z} Z G G G [21 ] makes it virtually compact special. The free group F ( X ) F(X) F ( X ) ∣ X ∣ + 1 |X| + 1 ∣ X ∣ + 1 [10 ] therefore makes G G G
For a hyperbolic virtually compact special group, Theorem 1.1 of [10 ] equates being virtually free-by-cyclic with a finite-index embedding in a free-by-cyclic group with finitely generated free kernel. In the rational cohomological dimension two branch of their proof the ambient quotient is Z \mathbb{Z} Z Z \mathbb{Z} Z Z \mathbb{Z} Z
Identify H H H Z \mathbb{Z} Z K = H ∩ F d K = H \cap F_d K = H ∩ F d Q ≤ Z Q \leq\mathbb{Z} Q ≤ Z F d F_d F d K K K
The companion also proves that
F ∗ ϕ = ⟨ F , t ∣ t − 1 f t = ϕ ( f ) ( f ∈ F ) ⟩ F_{\ast\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 virtually compact special whenever F F F ϕ : F → F \phi: F \to F ϕ : F → F F ∗ ϕ F_{\ast\phi} F ∗ ϕ [21 ] , Theorem 1.2. Such a group is finitely generated, so Kielak–Linton’s Corollary 1.8 [10 ] , followed by the same ambient argument, gives the finite-index embedding and subgroup-closed free-by-cyclic conclusion for these ascending HNN extensions as well. The resulting free kernel is not asserted to be their canonical height kernel.
