Introduction
Polyhedral products form a flexible construction connecting the combinatorics of simplicial complexes with homotopy theory, geometric group theory, and manifold topology. Given a simplicial complex L L L ( A , B ) = { ( A ( i ) , B ( i ) ) } i ∈ I (A,B)=\{(A(i),B(i))\}_{i\in I} ( A , B ) = {( A ( i ) , B ( i )) } i ∈ I Z L ( A , B ) Z_L(A,B) Z L ( A , B ) L L L A ( i ) A(i) A ( i ) B ( i ) B(i) B ( i ) [2 , 3 , 8 ] .
For aspherical polyhedral products, the interaction between the combinatorics of L L L relative graph product , determined by the actions G i = π 1 ( A ( i ) ) ↷ E i = π 0 ( π i − 1 ( B ( i ) ) ) G_i=\pi_1(A(i))\curvearrowright E_i=\pi_0(\pi^{-1}_i(B(i))) G i = π 1 ( A ( i )) ↷ E i = π 0 ( π i − 1 ( B ( i ))) p i : A ~ ( i ) → A ( i ) p_i:\widetilde{A}(i)\to A(i) p i : A ( i ) → A ( i ) H i ≤ G i H_i\leq G_i H i ≤ G i E i ≅ G i / H i E_i\cong G_i/H_i E i ≅ G i / H i Π L 1 ( G i , H i ) \Pi_{L}^{1}(G_i,H_i) Π L 1 ( G i , H i )
The purpose of this paper is to exploit this description in two directions. The first concerns invariants associated with families of subgroups, particularly amenable category and simplicial volume. The second concerns the Farrell–Jones conjecture and its rigidity consequences. A common feature of both directions is that global information about a graph product of pairs can be recovered from local information attached to the vertex pairs ( G i , H i ) (G_i,H_i) ( G i , H i )
Main results
The first result is a combination estimate for graph products of pairs. Let L L L G = Π L 1 ( G i , H i ) G=\Pi_{L}^{1}(G_i,H_i) G = Π L 1 ( G i , H i ) J ∈ S ( L ) J\in S(L) J ∈ S ( L )
G J = ∏ j ∈ J G j × ∏ i ∉ J H i , G ∅ = ∏ i ∈ I H i . G_J=\prod_{j\in J}G_j\times\prod_{i\notin J}H_i,\qquad G_{\varnothing}=\prod_{i\in I}H_i. G J = j ∈ J ∏ G j × i ∈ / J ∏ H i , G ∅ = i ∈ I ∏ H i . Theorem 1.1 (Combination theorem). Let F \mathcal{F} F G G G
cat F ( G ) ≤ max { cat F ∣ G ∅ ( G ∅ ) , max ∅ ≠ J ∈ S ( L ) ( gd ( G J ) + ∣ J ∣ ) } . \operatorname{cat}_{\mathcal{F}}(G)\leq\max\left\{\operatorname{cat}_{\mathcal{F}|G_{\varnothing}}(G_{\varnothing}),\ \max_{\varnothing\neq J\in S(L)}\left(\operatorname{gd}(G_J)+|J|\right)\right\}. cat F ( G ) ≤ max { cat F ∣ G ∅ ( G ∅ ) , ∅ = J ∈ S ( L ) max ( gd ( G J ) + ∣ J ∣ ) } . The same building-action argument applies more generally to relative graph products arising as fundamental groups of aspherical polyhedral products. In that setting, the relevant cell stabilizers are the residue stabilizers of the associated right-angled building; see Theorem 4.6 and the combination theorem of Capovilla–Li–Löh [7 ] .
The amenable case leads both to general upper bounds and, for split normal pairs, to exact computations.
Theorem 1.2 (Amenable category). Let L L L
If G = Π L 1 ( G i , H i ) G=\Pi_{L^1}(G_i,H_i) G = Π L 1 ( G i , H i ) G i G_i G i cat A m ( G ) ≤ dim L + 1 \operatorname{cat}_{\mathrm{Am}}(G)\leq\dim L+1 cat Am ( G ) ≤ dim L + 1
Suppose that G i = H i ⋊ Q i G_i=H_i\rtimes Q_i G i = H i ⋊ Q i i i i H i H_i H i cat A m Π L 1 ( G i , H i ) = cat A m ( ∗ L 1 Q i ) \operatorname{cat}_{\mathrm{Am}}\Pi_{L^1}(G_i,H_i)=\operatorname{cat}_{\mathrm{Am}}(\ast_{L^1}Q_i) cat Am Π L 1 ( G i , H i ) = cat Am ( ∗ L 1 Q i )
If, moreover, all Q i = Q Q_i=Q Q i = Q Q Q Q F F F cat A m Π L 1 ( G i , H i ) = vcd ( W L ) \operatorname{cat}_{\mathrm{Am}}\Pi_{L^1}(G_i,H_i)=\operatorname{vcd}(W_L) cat Am Π L 1 ( G i , H i ) = vcd ( W L )
As a consequence, whenever M M M m m m cat A m ( π 1 ( M ) ) < m \operatorname{cat}_{\mathrm{Am}}(\pi_1(M))<m cat Am ( π 1 ( M )) < m ∥ M ∥ = 0 \lVert M\rVert=0 ∥ M ∥ = 0 M M M
Our second main result is a Farrell–Jones combination theorem for graph products of pairs.
Theorem 1.3 (Farrell–Jones combination theorem). Let Γ \Gamma Γ { ( G v , H v ) } v ∈ V ( Γ ) \{(G_v,H_v)\}_{v\in V(\Gamma)} {( G v , H v ) } v ∈ V ( Γ ) v ∈ V ( Γ ) v\in V(\Gamma) v ∈ V ( Γ )
H v = { 1 } H_v=\{1\} H v = { 1 } G v ∈ C F J G_v\in\mathcal{C}_{\mathrm{FJ}} G v ∈ C FJ
H v ⊴ G v H_v\trianglelefteq G_v H v ⊴ G v G v / H v ∈ C F J G_v/H_v\in\mathcal{C}_{\mathrm{FJ}} G v / H v ∈ C FJ G v ∈ C F J G_v\in\mathcal{C}_{\mathrm{FJ}} G v ∈ C FJ H v H_v H v
G v = H v ∗ K v G_v=H_v\ast K_v G v = H v ∗ K v H v , K v ∈ C F J H_v,K_v\in\mathcal{C}_{\mathrm{FJ}} H v , K v ∈ C FJ
G v ∈ C F J G_v\in\mathcal{C}_{\mathrm{FJ}} G v ∈ C FJ ( G v , H v ) (G_v,H_v) ( G v , H v ) Proposition 7.19 .
Then Π Γ ( G v , H v ) ∈ C F J \Pi_\Gamma(G_v,H_v)\in\mathcal{C}_{\mathrm{FJ}} Π Γ ( G v , H v ) ∈ C FJ
The hypotheses are vertexwise: different vertices may be handled by different mechanisms. In particular, a single graph product of pairs may mix ordinary graph-product vertices, free-factor pairs, normal pairs controlled by cyclic-extension arguments, and non-normal pairs controlled by acylindrical Bass–Serre actions. Condition (ii) includes, for example, virtually solvable kernels and elementary amenable kernels of finite Hirsch length.
Corollary 1.4 (Borel rigidity). Let M = Z L ( A , B ) M=Z_L(A,B) M = Z L ( A , B ) G i ↷ E i G_i\curvearrowright E_i G i ↷ E i H i ≤ G i H_i\leq G_i H i ≤ G i E i ≅ G i / H i E_i\cong G_i/H_i E i ≅ G i / H i ( G i , H i ) (G_i,H_i) ( G i , H i ) Theorem 1.3 . Then M M M
The proofs of the two combination results use rather different mechanisms. For Theorem 1.1 , the key input is the action of the fundamental group on the standard realization of the associated right-angled building. The stabilizers are residue groups, and in the transitive case they are precisely the groups G J G_J G J
For Theorem 1.3 , we introduce a pair-deletion decomposition. Deleting a vertex v v v G v G_v G v H v H_v H v
( ∗ s ∈ S H v G v ) ⋊ Z . \left(\ast_{s\in S}^{H_v}G_v\right)\rtimes\mathbb{Z}. ( ∗ s ∈ S H v G v ) ⋊ Z . This leads to the local notion of FJ-admissibility and reduces the global Farrell–Jones problem to a condition on the individual vertex pairs.
The two main directions are therefore complementary. The F \mathcal{F} F Π L 1 ( G i , H i ) \Pi_{L^1}(G_i,H_i) Π L 1 ( G i , H i ) ( G i , H i ) (G_i,H_i) ( G i , H i )
The Farrell–Jones theorem also leads to topological rigidity applications. For closed aspherical manifold polyhedral products of dimension at least 5, the surgery-theoretic theorem of Bartels and Lück [4 ] converts the Farrell–Jones conclusion into the Borel conjecture. We illustrate this with real moment-angle manifolds and Möbius-type polyhedral products, and discuss surface-boundary pairs as a natural non-normal test case for the local theory.
Organization of the paper. In Section 2 we recall polyhedral products, the Davis–Kropholler asphericity criterion, and the associated right-angled building. Section 3 discusses relative graph products, graph products of pairs, the fundamental group of a polyhedral product, and the corresponding building action and cell stabilizers. We then develop the combination estimates for families of subgroups and apply them to amenable category and simplicial volume (Sections 4 and 5). Section 6 specializes the preceding constructions to polyhedral products of classifying-space pairs ( B G i , B H i ) (BG_i,BH_i) ( B G i , B H i )
Acknowledgements. The first author expresses his gratitude to P. Capovilla for helpful discussions and suggestions on earlier versions of this paper.
AI disclosure. All the ideas, arguments, results and proofs were obtained without the use of AI. We used ChatGPT for proofreading. The authors take full responsibility for the results.
Polyhedral products and asphericity
Polyhedral products
Let L L L I = { 1 , … , m } I=\{1,\ldots,m\} I = { 1 , … , m } S ( L ) S(L) S ( L ) L L L i ∈ I i\in I i ∈ I ( A ( i ) , B ( i ) ) (A(i),B(i)) ( A ( i ) , B ( i )) B ( i ) ⊆ A ( i ) B(i)\subseteq A(i) B ( i ) ⊆ A ( i )
Throughout, we assume that each A ( i ) A(i) A ( i ) B ( i ) B(i) B ( i )
Given a simplex J ∈ S ( L ) J\in S(L) J ∈ S ( L )
Z J ( A ‾ , B ‾ ) = ∏ i ∈ J A ( i ) × ∏ i ∈ I ∖ J B ( i ) . Z_J(\underline{A},\underline{B})=\prod_{i\in J}A(i)\times\prod_{i\in I\setminus J}B(i). Z J ( A , B ) = i ∈ J ∏ A ( i ) × i ∈ I ∖ J ∏ B ( i ) . where
( A ‾ , B ‾ ) = { ( A ( i ) , B ( i ) ) } i ∈ I . (\underline{A},\underline{B})=\{(A(i),B(i))\}_{i\in I}. ( A , B ) = {( A ( i ) , B ( i )) } i ∈ I . The polyhedral product associated with L L L ( A ‾ , B ‾ ) (\underline{A},\underline{B}) ( A , B )
Z L ( A ‾ , B ‾ ) = ⋃ J ∈ S ( L ) Z J ( A ‾ , B ‾ ) ⊆ ∏ i ∈ I A ( i ) . Z_L(\underline{A},\underline{B})=\bigcup_{J\in S(L)}Z_J(\underline{A},\underline{B})\subseteq\prod_{i\in I}A(i). Z L ( A , B ) = J ∈ S ( L ) ⋃ Z J ( A , B ) ⊆ i ∈ I ∏ A ( i ) . If all pairs ( A ( i ) , B ( i ) ) (A(i),B(i)) ( A ( i ) , B ( i )) ( A , B ) (A,B) ( A , B ) Z L ( A , B ) Z_L(A,B) Z L ( A , B ) [2 , 3 ] .
Equivalently, following Davis [8 ] , for a point x = ( x i ) i ∈ I ∈ ∏ i ∈ I A ( i ) x=(x_i)_{i\in I}\in\prod_{i\in I}A(i) x = ( x i ) i ∈ I ∈ ∏ i ∈ I A ( i )
Supp ( x ) = { i ∈ I ∣ x i ∈ A ( i ) ∖ B ( i ) } . \operatorname{Supp}(x)=\{i\in I\mid x_i\in A(i)\setminus B(i)\}. Supp ( x ) = { i ∈ I ∣ x i ∈ A ( i ) ∖ B ( i )} . Then
Z L ( A ‾ , B ‾ ) = { x ∈ ∏ i ∈ I A ( i ) ∣ Supp ( x ) ∈ S ( L ) } . Z_L(\underline{A},\underline{B})=\left\{x\in\prod_{i\in I}A(i)\mathrel{\Big|}\operatorname{Supp}(x)\in S(L)\right\}. Z L ( A , B ) = { x ∈ i ∈ I ∏ A ( i ) Supp ( x ) ∈ S ( L ) } . Thus the combinatorics of L L L B ( i ) B(i) B ( i )
The same construction may also be viewed categorically. Regard S ( L ) S(L) S ( L )
J ⟼ Z J ( A ‾ , B ‾ ) . J\longmapsto Z_J(\underline{A},\underline{B}). J ⟼ Z J ( A , B ) . Whenever J ⊆ J ′ J\subseteq J' J ⊆ J ′ B ( i ) ⊆ A ( i ) B(i)\subseteq A(i) B ( i ) ⊆ A ( i )
Z J ( A ‾ , B ‾ ) ↪ Z J ′ ( A ‾ , B ‾ ) . Z_J(\underline{A},\underline{B})\hookrightarrow Z_{J'}(\underline{A},\underline{B}). Z J ( A , B ) ↪ Z J ′ ( A , B ) . The polyhedral product is therefore the colimit
Z L ( A , B ) ≅ colim J ∈ S ( L ) Z J ( A , B ) . Z_L(A,B) \cong\operatorname{colim}_{J \in S(L)} Z_J(A,B). Z L ( A , B ) ≅ colim J ∈ S ( L ) Z J ( A , B ) . This point of view, emphasized in the general theory of polyhedral products, makes explicit that the space is assembled functorially from the face poset of L L L ( A ( i ) , B ( i ) ) (A(i),B(i)) ( A ( i ) , B ( i ))
Example 2.1. If ( A ( i ) , B ( i ) ) = ( [ 0 , 1 ] , { 1 } ) (A(i),B(i))=([0,1],\{1\}) ( A ( i ) , B ( i )) = ([ 0 , 1 ] , { 1 }) i ∈ I i\in I i ∈ I K ( L ) : = Z L ( [ 0 , 1 ] , { 1 } ) K(L):=Z_L([0,1],\{1\}) K ( L ) := Z L ([ 0 , 1 ] , { 1 }) chamber associated with L L L [9 Ex. A.4.8, A.5.2] ). The order complex of the face poset S ( L ) S(L) S ( L ) K ( L ) K(L) K ( L )
Example 2.2. For the pair ( D 1 , S 0 ) = ( [ − 1 , 1 ] , { − 1 , 1 } ) (D^1,S^0)=([-1,1],\{-1,1\}) ( D 1 , S 0 ) = ([ − 1 , 1 ] , { − 1 , 1 }) Z L R = Z L ( D 1 , S 0 ) Z_L^{\mathbb{R}}=Z_L(D^1,S^0) Z L R = Z L ( D 1 , S 0 ) real moment-angle complex . When L L L
Example 2.3. For ( D 2 , S 1 ) (D^2,S^1) ( D 2 , S 1 ) Z L = Z L ( D 2 , S 1 ) Z_L=Z_L(D^2,S^1) Z L = Z L ( D 2 , S 1 ) [6 ] ). Similarly, Z L ( D 4 , S 3 ) Z_L(D^4,S^3) Z L ( D 4 , S 3 ) [12 ] ).
Example 2.4. If G i G_i G i H i ≤ G i H_i\leq G_i H i ≤ G i ( B G i , B H i ) (BG_i,BH_i) ( B G i , B H i ) Z L ( B G , B H ) Z_L(BG,BH) Z L ( B G , B H )
Two features of the definition will be important throughout. First, the construction depends functorially both on the simplicial complex and on the collection of pairs. Second, although the definition uses all faces of L L L L L L L L L
Flag complexes and right-angularity
We now recall the combinatorial notions that underlie the asphericity theory of polyhedral products. Let L L L I I I L 1 L_1 L 1
Definition 2.5. A simplicial complex L L L flag if every finite subset J ⊆ I J\subseteq I J ⊆ I L L L L L L
J ∈ S ( L ) ⟺ { i , j } ∈ S ( L ) for every distinct i , j ∈ J . J\in S(L)\quad\Longleftrightarrow\quad\{i,j\}\in S(L)\ \text{for every distinct }i,j\in J. J ∈ S ( L ) ⟺ { i , j } ∈ S ( L ) for every distinct i , j ∈ J . Thus a flag complex is completely determined by its 1-skeleton. More precisely, given a simplicial graph Γ \Gamma Γ I I I flag completion , or clique complex , Flag ( Γ ) \operatorname{Flag}(\Gamma) Flag ( Γ ) Γ \Gamma Γ
L is flag ⟺ L = Flag ( L 1 ) . L\text{ is flag}\quad\Longleftrightarrow\quad L=\operatorname{Flag}(L_1). L is flag ⟺ L = Flag ( L 1 ) . Equivalently, L L L L L L J J J ∣ J ∣ |J| ∣ J ∣ dim J = ∣ J ∣ − 1 \dim J=|J|-1 dim J = ∣ J ∣ − 1 J J J ∣ J ∣ |J| ∣ J ∣ dim B L = dim L + 1 \dim B_L=\dim L+1 dim B L = dim L + 1
Example 2.6. If P P P Flag ( P ) \operatorname{Flag}(P) Flag ( P ) p 0 < p 1 < ⋯ < p k p_0<p_1<\cdots<p_k p 0 < p 1 < ⋯ < p k P P P P P P Flag ( P ) \operatorname{Flag}(P) Flag ( P )
The relevance of the flag condition to polyhedral products is already visible at the level of the 1-skeleton. The fundamental groups that arise from polyhedral products are governed by commutation relations associated with the edges of L L L L L L L L L missing faces which are invisible from L 1 L_1 L 1
We next recall the corresponding right-angled Coxeter group. Let Γ = L 1 \Gamma=L_1 Γ = L 1 right-angled Coxeter group associated with Γ \Gamma Γ
W Γ = ⟨ s i , i ∈ I ∣ s i 2 = 1 , [ s i , s j ] = 1 whenever { i , j } ∈ L 1 ⟩ . W_\Gamma=\langle s_i,\ i\in I\mid s_i^2=1,\ [s_i,s_j]=1\text{ whenever }\{i,j\}\in L_1\rangle. W Γ = ⟨ s i , i ∈ I ∣ s i 2 = 1 , [ s i , s j ] = 1 whenever { i , j } ∈ L 1 ⟩ . When the underlying simplicial complex is clear, we shall simply write W L W_L W L
For J ⊆ I J\subseteq I J ⊆ I W J = ⟨ s j ∣ j ∈ J ⟩ W_J=\langle s_j\mid j\in J\rangle W J = ⟨ s j ∣ j ∈ J ⟩ J J J L L L J J J W J ≅ ( Z / 2 ) ∣ J ∣ W_J\cong(\mathbb{Z}/2)^{|J|} W J ≅ ( Z /2 ) ∣ J ∣
Conversely, in a right-angled Coxeter group, W J W_J W J J J J L 1 L_1 L 1 L L L
J ∈ S ( L ) ⟺ W J is finite . J\in S(L)\Longleftrightarrow W_J\text{ is finite}. J ∈ S ( L ) ⟺ W J is finite . In the terminology of Coxeter groups, such subsets J J J spherical .
This correspondence identifies the face poset S ( L ) S(L) S ( L ) W L W_L W L
Asphericity criteria of polyhedral products
We next recall the asphericity criterion for polyhedral products. The original criterion of M.Davis [8 ] was subsequently refined by M.Davis and Kropholler [11 ] ; in particular, the corrected statement distinguishes those coordinates which contribute nontrivially to the associated right-angled structure. We formulate the corrected criterion below.
Throughout this subsection, let
Z L ( A , B ) = Z L ( ( A ( i ) , B ( i ) ) i ∈ I ) Z_L(A,B)=Z_L\bigl((A(i),B(i))_{i\in I}\bigr) Z L ( A , B ) = Z L ( ( A ( i ) , B ( i ) ) i ∈ I ) be a polyhedral product, where the A ( i ) A(i) A ( i ) B ( i ) B(i) B ( i ) A ( i ) A(i) A ( i ) ∗ i ∈ B ( i ) *_{i}\in B(i) ∗ i ∈ B ( i ) i ∈ I i\in I i ∈ I p i : A ~ ( i ) ⟶ A ( i ) p_i:\widetilde{A}(i)\longrightarrow A(i) p i : A ( i ) ⟶ A ( i ) B ~ ( i ) = p i − 1 ( B ( i ) ) \widetilde{B}(i)=p_i^{-1}(B(i)) B ( i ) = p i − 1 ( B ( i )) G i = π 1 ( A ( i ) , ∗ i ) G_i=\pi_1(A(i),*_{i}) G i = π 1 ( A ( i ) , ∗ i )
E i : = π 0 ( B ~ ( i ) ) . E_i:=\pi_0\bigl(\widetilde{B}(i)\bigr). E i := π 0 ( B ( i ) ) . Equivalently, E i E_i E i π 1 \pi_1 π 1 π 1 ( A ( i ) , B ( i ) , ∗ i ) \pi_1(A(i),B(i),*_{i}) π 1 ( A ( i ) , B ( i ) , ∗ i ) [11 ] , define I ′ ′ = { i ∈ I ∣ ∣ E i ∣ ≠ 1 } I^{\prime\prime}=\{i\in I\mid|E_i|\ne1\} I ′′ = { i ∈ I ∣ ∣ E i ∣ = 1 } L ′ ′ = L I ′ ′ L^{\prime\prime}=L_{I^{\prime\prime}} L ′′ = L I ′′ L L L I ′ ′ I^{\prime\prime} I ′′
Definition 2.7. A vertex i i i K K K conelike if it is joined by an edge to every other vertex of K K K
Thus, if i i i K K K K = { i } ∗ K V ( K ) ∖ { i } K=\{i\}*K_{V(K)\setminus\{i\}} K = { i } ∗ K V ( K ) ∖ { i }
Definition 2.8. Let ( A , B ) (A,B) ( A , B ) A A A A ~ ⟶ A \widetilde{A}\longrightarrow A A ⟶ A B ~ \widetilde{B} B B B B
The pair ( A , B ) (A,B) ( A , B ) aspherical if A ~ \widetilde{A} A B ~ \widetilde{B} B
The pair ( A , B ) (A,B) ( A , B ) nearly aspherical if A ~ \widetilde{A} A B ~ \widetilde{B} B
In particular, if each path component of B B B
π 1 ( B , b ) ⟶ π 1 ( A , b ) , \pi_{1}(B,b) \longrightarrow\pi_{1}(A,b), π 1 ( B , b ) ⟶ π 1 ( A , b ) , then ( A , B ) (A,B) ( A , B )
To state the general criterion, one further decomposition of the vertex set is required. Fix a field F \mathbb{F} F I ∖ I ′ ′ I \setminus I^{\prime\prime} I ∖ I ′′
I 1 = { i ∈ I ∖ I ′ ′ ∣ B ~ ( i ) is acyclic over Z } , I_{1}=\left\{i\in I\setminus I^{\prime\prime}\mid\widetilde{B}(i)\text{ is acyclic over }\mathbb{Z}\right\}, I 1 = { i ∈ I ∖ I ′′ ∣ B ( i ) is acyclic over Z } , and
I 2 = { i ∈ I ∖ I ′ ′ ∣ B ~ ( i ) is not acyclic over F } . I_{2}=\left\{i\in I\setminus I^{\prime\prime}\mid\widetilde{B}(i)\text{ is not acyclic over }\mathbb{F}\right\}. I 2 = { i ∈ I ∖ I ′′ ∣ B ( i ) is not acyclic over F } . We shall write
I ′ = I 2 ∪ I ′ ′ , L ′ = L I ′ . I^{\prime}=I_{2}\cup I^{\prime\prime},\qquad L^{\prime}=L_{I^{\prime}}. I ′ = I 2 ∪ I ′′ , L ′ = L I ′ . Following [11 ] , we impose the auxiliary condition
I 1 ⊔ I 2 = I ∖ I ′ ′ (1) I_{1}\sqcup I_{2}=I\setminus I^{\prime\prime} \tag*{(1)} I 1 ⊔ I 2 = I ∖ I ′′ (1) for some field F \mathbb{F} F B ~ ( i ) \widetilde{B}(i) B ( i )
Theorem 2.9 (Davis–Kropholler). Assume that (2.1) holds. Then the polyhedral product Z L ( A , B ) Z_{L}(A,B) Z L ( A , B )
(i) Each A ( i ) A(i) A ( i )
(ii) If i ∈ I ′ i\in I^{\prime} i ∈ I ′ L ′ L^{\prime} L ′ ( A ( i ) , B ( i ) ) (A(i),B(i)) ( A ( i ) , B ( i ))
(iii) Every vertex of I 2 I_{2} I 2 L ′ L^{\prime} L ′ ⟨ I 2 ⟩ \langle I_{2}\rangle ⟨ I 2 ⟩
L ′ = ⟨ I 2 ⟩ ∗ L ′ ′ . L^{\prime}=\langle I_{2}\rangle* L^{\prime\prime}. L ′ = ⟨ I 2 ⟩ ∗ L ′′ . (iv) The simplicial complex L ′ L^{\prime} L ′
Remark 2.10. The distinction between L L L L ′ L^{\prime} L ′ not require the whole complex L L L ∣ E i ∣ = 1 |E_{i}|=1 ∣ E i ∣ = 1 L ′ L^{\prime} L ′ L L L
For the class of pairs most relevant to the sequel, the criterion simplifies considerably. Suppose that, for every i ∈ I i\in I i ∈ I C ⊆ B ( i ) C\subseteq B(i) C ⊆ B ( i )
π 1 ( C ) ⟶ π 1 ( A ( i ) ) , \pi_{1}(C)\longrightarrow\pi_{1}(A(i)), π 1 ( C ) ⟶ π 1 ( A ( i )) , and suppose moreover that whenever B ( i ) B(i) B ( i ) π 1 ( A ( i ) ) \pi_{1}(A(i)) π 1 ( A ( i ))
∣ E i ∣ ≥ 2 for every i ∈ I , |E_{i}|\geq2\qquad\text{for every }i\in I, ∣ E i ∣ ≥ 2 for every i ∈ I , so that
I ′ ′ = I , L ′ ′ = L , I 1 = I 2 = ∅ . I^{\prime\prime}=I,\qquad L^{\prime\prime}=L,\qquad I_{1}=I_{2}=\varnothing. I ′′ = I , L ′′ = L , I 1 = I 2 = ∅ . The corrected theorem therefore reduces to the following useful form.
Corollary 2.11. Assume that for every i ∈ I i\in I i ∈ I
(1) each path component of B ( i ) B(i) B ( i ) π 1 \pi_{1} π 1 A ( i ) A(i) A ( i )
(2) if B ( i ) B(i) B ( i ) π 1 ( A ( i ) ) \pi_{1}(A(i)) π 1 ( A ( i ))
Then Z L ( A , B ) Z_{L}(A,B) Z L ( A , B )
(1) each A ( i ) A(i) A ( i )
(2) for every nonconelike vertex i i i L L L ( A ( i ) , B ( i ) ) (A(i),B(i)) ( A ( i ) , B ( i )) (3) L L L
Proof. Under the assumptions above, every E i E_i E i I ′ ′ = I I^{\prime\prime}=I I ′′ = I L ′ ′ = L L^{\prime\prime}=L L ′′ = L I 1 = I 2 = ∅ I_1=I_2=\varnothing I 1 = I 2 = ∅ Theorem 2.9 reduces to conditions (i), (ii), and the flagness of L L L π 1 \pi_1 π 1 B ( i ) B(i) B ( i ) B ~ ( i ) \widetilde{B}(i) B ( i ) B ( i ) B(i) B ( i )
The role of the flag condition can be seen geometrically after passing to universal covers. Once the spaces A ( i ) A(i) A ( i ) [8 ] reduces the relevant homotopy-theoretic problem to polyhedral products of pairs ( Cone ( E i ) , E i ) (\operatorname{Cone}(E_i),E_i) ( Cone ( E i ) , E i ) E i E_i E i B ~ ( i ) \widetilde{B}(i) B ( i ) ∣ E i ∣ = 1 |E_i|=1 ∣ E i ∣ = 1
This geometric model will be recalled in Subsection 2.4. It will later provide the contractible G G G G = π 1 Z L ( A , B ) G=\pi_1 Z_L(A,B) G = π 1 Z L ( A , B )
The basic construction and right-angled buildings
We next recall the basic construction associated with a mirrored space and explain its relation with right-angled Coxeter groups and right-angled buildings. This construction provides the geometric model that will later be used to study the action of the fundamental group of an aspherical polyhedral product.
Let I I I I I I X X X { X i } i ∈ I \{X_i\}_{i\in I} { X i } i ∈ I x ∈ X x\in X x ∈ X I ( x ) = { i ∈ I ∣ x ∈ X i } I(x)=\{i\in I\mid x\in X_i\} I ( x ) = { i ∈ I ∣ x ∈ X i } J ⊆ I J\subseteq I J ⊆ I X J = ⋂ j ∈ J X j X_J=\bigcap_{j\in J}X_j X J = ⋂ j ∈ J X j X ∅ = X X_\varnothing=X X ∅ = X
Let ( W , S ) (W,S) ( W , S ) I I I S = { s i } i ∈ I S=\{s_i\}_{i\in I} S = { s i } i ∈ I J ⊆ I J\subseteq I J ⊆ I W J = ⟨ s j ∣ j ∈ J ⟩ W_J=\langle s_j\mid j\in J\rangle W J = ⟨ s j ∣ j ∈ J ⟩ basic construction associated with ( W , X ) (W,X) ( W , X ) U ( W , X ) = ( W × X ) / ∼ \mathcal{U}(W,X)=(W\times X)/\sim U ( W , X ) = ( W × X ) / ∼ ( w , x ) ∼ ( w ′ , x ′ ) (w,x)\sim(w',x') ( w , x ) ∼ ( w ′ , x ′ ) x = x ′ x=x' x = x ′ w W I ( x ) = w ′ W I ( x ) wW_{I(x)}=w'W_{I(x)} w W I ( x ) = w ′ W I ( x )
( w , x ) ∼ ( w ′ , x ) ⟺ w − 1 w ′ ∈ W I ( x ) . (w,x)\sim(w',x)\quad\Longleftrightarrow\quad w^{-1}w'\in W_{I(x)}. ( w , x ) ∼ ( w ′ , x ) ⟺ w − 1 w ′ ∈ W I ( x ) . The group W W W U ( W , X ) \mathcal{U}(W,X) U ( W , X ) g ⋅ [ w , x ] = [ g w , x ] g\cdot[w,x]=[gw,x] g ⋅ [ w , x ] = [ g w , x ] { w } × X \{w\}\times X { w } × X w X wX w X U ( W , X ) \mathcal{U}(W,X) U ( W , X ) X X X W J W_J W J
We apply this construction to the chamber determined by a simplicial complex. Let L L L I I I S ( L ) S(L) S ( L ) K ( L ) = ∣ Flag ( S ( L ) ) ∣ K(L)=|\operatorname{Flag}(S(L))| K ( L ) = ∣ Flag ( S ( L )) ∣ K ( L ) K(L) K ( L ) I I I i ∈ I i\in I i ∈ I i i i
K i = ∣ Flag { J ∈ S ( L ) ∣ i ∈ J } ∣ . K_i=|\operatorname{Flag}\{J\in S(L)\mid i\in J\}|. K i = ∣ Flag { J ∈ S ( L ) ∣ i ∈ J } ∣. and for a simplex J ∈ S ( L ) J\in S(L) J ∈ S ( L )
K J = ⋂ j ∈ J K j = ∣ Flag ( S ( L ) ≥ J ) ∣ . K_J=\bigcap_{j\in J}K_j=|\operatorname{Flag}(S(L)_{\geq J})|. K J = j ∈ J ⋂ K j = ∣ Flag ( S ( L ) ≥ J ) ∣. where
S ( L ) ≥ J = { J ′ ∈ S ( L ) ∣ J ⊆ J ′ } . S(L)_{\geq J}=\{J'\in S(L)\mid J\subseteq J'\}. S ( L ) ≥ J = { J ′ ∈ S ( L ) ∣ J ⊆ J ′ } . Hence the intersections of mirrors are indexed by the simplices of L L L
Let W L W_L W L 1 1 1 L L L
W L = ⟨ s i , i ∈ I | s i 2 = 1 , [ s i , s j ] = 1 whenever { i , j } ∈ L 1 ⟩ . W_L=\left\langle s_i,\ i\in I\ \middle|\ s_i^2=1,\ [s_i,s_j]=1\ \text{whenever }\{i,j\}\in L_1\right\rangle. W L = ⟨ s i , i ∈ I s i 2 = 1 , [ s i , s j ] = 1 whenever { i , j } ∈ L 1 ⟩ . The associated basic construction Σ L : = U ( W L , K ( L ) ) \Sigma_L := \mathcal{U}(W_L, K(L)) Σ L := U ( W L , K ( L )) L L L Σ L \Sigma_L Σ L W L W_L W L
The cells of Σ L \Sigma_L Σ L J 0 ⊊ J 1 ⊊ ⋯ ⊊ J k J_0 \subsetneq J_1 \subsetneq\cdots\subsetneq J_k J 0 ⊊ J 1 ⊊ ⋯ ⊊ J k S ( L ) S(L) S ( L ) J ∈ S ( L ) J \in S(L) J ∈ S ( L ) w W J wW_J w W J W L W_L W L w W J w − 1 wW_Jw^{-1} w W J w − 1 L L L
When L L L Σ L \Sigma_L Σ L L L L
L flag ⟹ U ( W L , K ( L ) ) is contractible . L\text{ flag} \Longrightarrow\mathcal{U}(W_L,K(L))\text{ is contractible}. L flag ⟹ U ( W L , K ( L )) is contractible . The Coxeter complex above should be viewed as an apartment in a more general right-angled building. We recall the form needed below. A chamber system over I I I is a set C C C ∼ i \sim_i ∼ i i ∈ I i \in I i ∈ I ∼ i \sim_i ∼ i i i i J ⊆ I J \subseteq I J ⊆ I ∼ j \sim_j ∼ j j ∈ J j \in J j ∈ J J J J
Given such a chamber system C C C I I I X X X U ( C , X ) = ( C × X ) / ∼ \mathcal{U}(C,X) = (C \times X)/\sim U ( C , X ) = ( C × X ) / ∼ ( c , x ) ∼ ( c ′ , x ) (c,x) \sim(c',x) ( c , x ) ∼ ( c ′ , x ) c c c c ′ c' c ′ I ( x ) I(x) I ( x ) { c } × X \{c\} \times X { c } × X
When C C C ( W L , S ) (W_L,S) ( W L , S ) X = K ( L ) X = K(L) X = K ( L ) U ( C , K ( L ) ) \mathcal{U}(C,K(L)) U ( C , K ( L )) standard realization of the building. Each apartment is naturally isomorphic to U ( W L , K ( L ) ) \mathcal{U}(W_L,K(L)) U ( W L , K ( L )) L L L
We now relate this construction to polyhedral products. Let E i E_i E i i ∈ I i \in I i ∈ I ( Cone ( E i ) , E i ) (\operatorname{Cone}(E_i),E_i) ( Cone ( E i ) , E i ) Z L ( Cone ( E ) , E ) Z_L(\operatorname{Cone}(E),E) Z L ( Cone ( E ) , E ) ∏ i ∈ I E i \prod_{i\in I} E_i ∏ i ∈ I E i U ( C , K ( L ) ) \mathcal{U}(C,K(L)) U ( C , K ( L )) C C C ( W L , S ) (W_L,S) ( W L , S ) E i E_i E i
In particular, in the case ∣ E i ∣ > 1 |E_i|>1 ∣ E i ∣ > 1 i ∈ I i \in I i ∈ I L L L [8 Proposition 2.10 and Lemma 2.11] )
Z L ( Cone ( E ) , E ) ≃ U ( C , K ( L ) ) . Z_L(\operatorname{Cone}(E),E) \simeq\mathcal{U}(C,K(L)). Z L ( Cone ( E ) , E ) ≃ U ( C , K ( L )) . This is the geometric model which enters Davis’s proof of the asphericity theorem.
More generally, if some E i E_i E i
L ′ ′ = L I ′ ′ , I ′ ′ = { i ∈ I ∣ ∣ E i ∣ ≠ 1 } . L'' = L_{I''}, \qquad I'' = \{i \in I \mid|E_i| \ne1\}. L ′′ = L I ′′ , I ′′ = { i ∈ I ∣ ∣ E i ∣ = 1 } . The Davis–Kropholler correction [11 ] shows that the relevant building is then the right-angled building associated with L ′ ′ L'' L ′′ L ′ ′ L'' L ′′ L L L
The importance of this construction for the sequel is that it converts the combinatorics of a simplicial complex into a contractible equivariant cell complex. For an aspherical polyhedral product Z L ( A , B ) Z_L(A,B) Z L ( A , B ) G = π 1 Z L ( A , B ) G = \pi_1 Z_L(A,B) G = π 1 Z L ( A , B ) G G G L L L [7 ] .
Fundamental groups and relative graph products
Relative graph products
Relative graph products. We now recall the relative graph-product construction used by Davis to describe fundamental groups of polyhedral products. The ordinary graph product of groups is recovered as a special case, but the relative construction allows the local groups to act nontrivially on sets and is therefore adapted to the general pairs appearing in Z L ( A , B ) Z_L(A,B) Z L ( A , B )
Let Γ \Gamma Γ I I I i ∈ I i \in I i ∈ I G i G_i G i E i E_i E i N i = ker ( G i ↷ E i ) N_i = \ker(G_i \curvearrowright E_i) N i = ker ( G i ↷ E i ) G ‾ i = G i / N i \overline{G}_i = G_i/N_i G i = G i / N i E i E_i E i
We first recall the ordinary graph product. The graph product of the family ( G i ) i ∈ I (G_i)_{i\in I} ( G i ) i ∈ I Γ \Gamma Γ
Γ ∗ G i = ( i ∈ I ∗ G i ) / ⟨ ⟨ [ g i , g j ] ∣ g i ∈ G i , g j ∈ G j , { i , j } ∈ E ( Γ ) ⟩ ⟩ . {}^{*}_{\Gamma}G_i=\left({}^{*}_{i\in I}G_i\right)\Big/\Big\langle\!\Big\langle[g_i,g_j]\mid g_i\in G_i,\ g_j\in G_j,\ \{i,j\}\in E(\Gamma)\Big\rangle\!\Big\rangle. Γ ∗ G i = ( i ∈ I ∗ G i ) / ⟨ ⟨ [ g i , g j ] ∣ g i ∈ G i , g j ∈ G j , { i , j } ∈ E ( Γ ) ⟩ ⟩ . Thus elements belonging to vertex groups associated with adjacent vertices are required to commute.
For a complete subgraph J ⊆ I J \subseteq I J ⊆ I { G j } j ∈ J \{G_j\}_{j\in J} { G j } j ∈ J G J = ∏ j ∈ J G j G_J=\prod_{j\in J}G_j G J = ∏ j ∈ J G j L L L Γ = L ( 1 ) \Gamma=L^{(1)} Γ = L ( 1 ) Γ \Gamma Γ L L L L L L
The relative graph product is defined from the permutation data ( G i , E i ) (G_i,E_i) ( G i , E i ) G = ∏ i ∈ I G i G=\prod_{i\in I}G_i G = ∏ i ∈ I G i N = ∏ i ∈ I N i N=\prod_{i\in I}N_i N = ∏ i ∈ I N i N i = ker ( G i ↷ E i ) N_i=\ker(G_i\curvearrowright E_i) N i = ker ( G i ↷ E i ) G G G Z Γ ( Cone ( E ) , E ) Z_\Gamma(\operatorname{Cone}(E),E) Z Γ ( Cone ( E ) , E ) N N N G / N G/N G / N
Let Z Γ ( Cone ( E ) , E ) Z_\Gamma(\operatorname{Cone}(E),E) Z Γ ( Cone ( E ) , E ) Γ Γ \Gamma_\Gamma Γ Γ G / N G/N G / N
1 ⟶ π 1 Z Γ ( Cone ( E ) , E ) ⟶ Γ Γ ⟶ G / N ⟶ 1. 1\longrightarrow\pi_1Z_\Gamma(\operatorname{Cone}(E),E)\longrightarrow\Gamma_\Gamma\longrightarrow G/N\longrightarrow1. 1 ⟶ π 1 Z Γ ( Cone ( E ) , E ) ⟶ Γ Γ ⟶ G / N ⟶ 1. Davis proves that Γ Γ \Gamma_\Gamma Γ Γ
The relative graph product Z Γ ( G , E ) Z_\Gamma(G,E) Z Γ ( G , E )
Z Γ ( G , E ) → Γ Γ ↓ ↓ G → G / N . \begin{CD}
Z_\Gamma(G,E) @>>> \Gamma_\Gamma\\
@VVV @VVV \\
G @>>> G/N.
\end{CD} Z Γ ( G , E ) ↓ ⏐ G Γ Γ ↓ ⏐ G / N . Consequently, it fits into an exact sequence
1 ⟶ N ⟶ Z Γ ( G , E ) ⟶ Γ Γ ⟶ 1. 1\longrightarrow N\longrightarrow Z_\Gamma(G,E)\longrightarrow\Gamma_\Gamma\longrightarrow1. 1 ⟶ N ⟶ Z Γ ( G , E ) ⟶ Γ Γ ⟶ 1. The construction interpolates between several familiar cases.
Example 3.1. If each action G i ↷ E i G_i\curvearrowright E_i G i ↷ E i N i = 1 N_i=1 N i = 1 i i i E i E_i E i
Example 3.2. If every E i E_i E i N i = G i N_i=G_i N i = G i
Example 3.3. The ordinary graph product is recovered when the local permutation data are chosen so that no additional kernel correction is present. In particular, in the standard classifying-space case ( B G i , ∗ ) (BG_i,*) ( B G i , ∗ ) G i G_i G i
For a subset J ⊆ I J\subseteq I J ⊆ I Z J ( G , E ) Z_J(G,E) Z J ( G , E ) J J J J J J Z J ( G , E ) ≅ ∏ j ∈ J G j Z_J(G,E)\cong\prod_{j\in J}G_j Z J ( G , E ) ≅ ∏ j ∈ J G j
This local notation will be useful later when we describe stabilizers of cells in the right-angled building associated with an aspherical polyhedral product.
A particularly important case occurs when each action G i ↷ E i G_i \curvearrowright E_i G i ↷ E i e i ∈ E i e_i \in E_i e i ∈ E i H i = Stab G i ( e i ) H_i = \operatorname{Stab}_{G_i}(e_i) H i = Stab G i ( e i ) E i ≅ G i / H i E_i \cong G_i/H_i E i ≅ G i / H i [8 Remark 2.15] that the relative graph product Z L 1 ( G , E ) Z_{L_1}(\mathbf{G},\mathbf{E}) Z L 1 ( G , E ) Π L 1 ( G i , H i ) \Pi_{L_1}(G_i,H_i) Π L 1 ( G i , H i ) H i = { 1 } H_i = \{1\} H i = { 1 } i i i
Definition 3.4. Let Γ \Gamma Γ I I I H i ≤ G i H_i \leq G_i H i ≤ G i i ∈ I i \in I i ∈ I graph product of pairs Π Γ ( G i , H i ) \Pi_\Gamma(G_i,H_i) Π Γ ( G i , H i )
G J = ∏ j ∈ J G j × ∏ i ∉ J H i , G_J = \prod_{j \in J} G_j \times\prod_{i \notin J} H_i, G J = j ∈ J ∏ G j × i ∈ / J ∏ H i , indexed by the cliques J J J Γ \Gamma Γ
Finally, we emphasize that the relative graph product depends only on the graph Γ \Gamma Γ L 1 L_1 L 1 L L L
The fundamental group of a polyhedral product
We now relate the topology of a polyhedral product to the relative graph-product construction introduced in the previous subsection.
Let ( A , B ) = { ( A ( i ) , B ( i ) ) } i ∈ I (A,B) = \{(A(i),B(i))\}_{i\in I} ( A , B ) = {( A ( i ) , B ( i )) } i ∈ I I I I L L L A ( i ) A(i) A ( i ) i ∈ I i \in I i ∈ I p i : A ~ ( i ) ⟶ A ( i ) p_i\colon\widetilde{A}(i) \longrightarrow A(i) p i : A ( i ) ⟶ A ( i ) B ~ ( i ) = p i − 1 ( B ( i ) ) \widetilde{B}(i) = p_i^{-1}(B(i)) B ( i ) = p i − 1 ( B ( i )) G i = π 1 ( A ( i ) ) G_i = \pi_1(A(i)) G i = π 1 ( A ( i )) G i G_i G i A ~ ( i ) \widetilde{A}(i) A ( i ) B ~ ( i ) \widetilde{B}(i) B ( i ) G i G_i G i E i = π 0 ( B ~ ( i ) ) E_i = \pi_0(\widetilde{B}(i)) E i = π 0 ( B ( i ))
The collection ( G , E ) = { ( G i , E i ) } i ∈ I (\mathbf{G},\mathbf{E}) = \{(G_i,E_i)\}_{i\in I} ( G , E ) = {( G i , E i ) } i ∈ I fundamental-group data associated with ( A , B ) (A,B) ( A , B )
The first observation is that the fundamental group of a polyhedral product is insensitive to the simplices of dimension greater than one.
Lemma 3.5 (M.Davis cf. [8 Lemma 2.17] ). Let L 1 L_1 L 1 L L L Z L 1 ( A , B ) ⟶ Z L ( A , B ) Z_{L_1}(A,B) \longrightarrow Z_L(A,B) Z L 1 ( A , B ) ⟶ Z L ( A , B )
π 1 ( Z L 1 ( A , B ) ) → ≅ π 1 ( Z L ( A , B ) ) . \pi_1\bigl(Z_{L_1}(A,B)\bigr) \xrightarrow{\cong} \pi_1\bigl(Z_L(A,B)\bigr). π 1 ( Z L 1 ( A , B ) ) ≅ π 1 ( Z L ( A , B ) ) . In particular, π 1 ( Z L ( A , B ) ) \pi_1\bigl(Z_L(A,B)\bigr) π 1 ( Z L ( A , B ) ) L 1 L_1 L 1
Remark 3.6. This fact is conceptually important for what follows. The fundamental group is determined entirely by the graph L 1 L_1 L 1 L L L L = Flag ( L 1 ) L = \operatorname{Flag}(L_1) L = Flag ( L 1 )
The fundamental group can now be described explicitly in terms of the relative graph product of the local data.
Theorem 3.7 (M.Davis cf. [8 Theorem 2.18] ). Let L L L I I I ( A , B ) (A,B) ( A , B ) ( G , E ) = { ( G i , E i ) } i ∈ I (\mathbf{G},\mathbf{E}) = \{(G_i,E_i)\}_{i\in I} ( G , E ) = {( G i , E i ) } i ∈ I
G i = π 1 ( A ( i ) ) , E i = π 0 ( B ~ ( i ) ) . G_i = \pi_1(A(i)), \qquad E_i = \pi_0(\widetilde{B}(i)). G i = π 1 ( A ( i )) , E i = π 0 ( B ( i )) . Then there is a natural isomorphism π 1 ( Z L ( A , B ) ) ≅ Z L 1 ( G , E ) \pi_1\bigl(Z_L(A,B)\bigr) \cong Z_{L_1}(\mathbf{G},\mathbf{E}) π 1 ( Z L ( A , B ) ) ≅ Z L 1 ( G , E ) L 1 L_1 L 1
Proof. For each i ∈ I i \in I i ∈ I B ~ ( i ) \widetilde{B}(i) B ( i ) G i G_i G i B ~ ( i ) ⟶ E i \widetilde{B}(i) \longrightarrow E_i B ( i ) ⟶ E i A ~ ( i ) \widetilde{A}(i) A ( i ) G i G_i G i
q i : A ~ ( i ) ⟶ Cone ( E i ) . q_i:\widetilde{A}(i)\longrightarrow\operatorname{Cone}(E_i). q i : A ( i ) ⟶ Cone ( E i ) . Taking polyhedral products yields a map
q : Z L ( A ~ , B ~ ) ⟶ Z L ( Cone ( E ) , E ) . q:Z_L(\widetilde{A},\widetilde{B})\longrightarrow Z_L(\operatorname{Cone}(E),E). q : Z L ( A , B ) ⟶ Z L ( Cone ( E ) , E ) . The universal covering of Z L ( A ~ , B ~ ) Z_L(\widetilde{A},\widetilde{B}) Z L ( A , B ) q q q Z L ( Cone ( E ) , E ) Z_L(\operatorname{Cone}(E),E) Z L ( Cone ( E ) , E ) Z ~ L ( A ~ , B ~ ) \widetilde{Z}_L(\widetilde{A},\widetilde{B}) Z L ( A , B ) Z ~ L ( Cone ( E ) , E ) \widetilde{Z}_L(\operatorname{Cone}(E),E) Z L ( Cone ( E ) , E ) π 1 Z L ( A , B ) ⟶ Z L 1 ( G , E ) \pi_1 Z_L(A,B) \longrightarrow Z_{L_1}(G,E) π 1 Z L ( A , B ) ⟶ Z L 1 ( G , E )
By the relative graph-product construction recalled in Subsection 3.1, this group of lifts is precisely Z L 1 ( G , E ) Z_{L_1}(G,E) Z L 1 ( G , E ) Z L ( A , B ) Z_L(A,B) Z L ( A , B )
π 1 ( Z L ( A , B ) ) ≅ Z L 1 ( G , E ) . \pi_1(Z_L(A,B)) \cong Z_{L_1}(G,E). π 1 ( Z L ( A , B )) ≅ Z L 1 ( G , E ) . Example 3.8 (Graph products of groups). Let ( A ( i ) , B ( i ) ) = ( B G i , ∗ ) (A(i),B(i))=(BG_i,\ast) ( A ( i ) , B ( i )) = ( B G i , ∗ ) A ~ ( i ) = E G i \widetilde{A}(i)=EG_i A ( i ) = E G i G i G_i G i E i ≅ G i E_i\cong G_i E i ≅ G i G i G_i G i Theorem 3.7 therefore recovers the ordinary graph product: π 1 Z L ( B G , ∗ ) ≅ ∗ L 1 G i \pi_1 Z_L(BG,\ast)\cong\ast_{L_1}G_i π 1 Z L ( B G , ∗ ) ≅ ∗ L 1 G i
Example 3.9 (Right-angled Artin groups). If ( A ( i ) , B ( i ) ) = ( S 1 , ∗ ) (A(i),B(i))=(S^1,\ast) ( A ( i ) , B ( i )) = ( S 1 , ∗ ) i ∈ I i\in I i ∈ I G i ≅ Z G_i\cong\mathbb{Z} G i ≅ Z π 1 Z L ( S 1 , ∗ ) ≅ A L 1 \pi_1 Z_L(S^1,\ast)\cong A_{L_1} π 1 Z L ( S 1 , ∗ ) ≅ A L 1 L 1 L_1 L 1
Example 3.10 (Right-angled Coxeter groups). Let ( A ( i ) , B ( i ) ) = ( B Z / 2 , ∗ ) (A(i),B(i))=(B\mathbb{Z}/2,\ast) ( A ( i ) , B ( i )) = ( B Z /2 , ∗ ) π 1 Z L ( B Z / 2 , ∗ ) ≅ W L 1 \pi_1 Z_L(B\mathbb{Z}/2,\ast)\cong W_{L_1} π 1 Z L ( B Z /2 , ∗ ) ≅ W L 1 L 1 L_1 L 1
Example 3.11 (Nontrivial relative data). Suppose that each B ( i ) B(i) B ( i ) A ~ ( i ) \widetilde{A}(i) A ( i ) G i ↷ E i G_i\curvearrowright E_i G i ↷ E i N i = ker ( G i ↷ E i ) N_i=\ker(G_i\curvearrowright E_i) N i = ker ( G i ↷ E i ) G i G_i G i π 1 Z L ( A , B ) ≅ Z L 1 ( G , E ) \pi_1 Z_L(A,B)\cong Z_{L_1}(G,E) π 1 Z L ( A , B ) ≅ Z L 1 ( G , E ) N i N_i N i
Remark 3.12. The relative graph-product construction can be nontrivial even when all local groups G i G_i G i G i G_i G i E i E_i E i Z L 1 ( G , E ) ≅ π 1 Z L 1 ( D 1 , S 0 ) Z_{L_1}(G,E)\cong\pi_1 Z_{L_1}(D^1,S^0) Z L 1 ( G , E ) ≅ π 1 Z L 1 ( D 1 , S 0 )
Theorem 3.7 identifies the acting group that will be relevant throughout the sequel. Under the asphericity hypotheses of Subsection 2.3, we shall write G = π 1 Z L ( A , B ) ≅ Z L 1 ( G , E ) G=\pi_1 Z_L(A,B)\cong Z_{L_1}(G,E) G = π 1 Z L ( A , B ) ≅ Z L 1 ( G , E ) G G G
The building action and cell stabilizers
We now combine the relative graph-product description of the fundamental group with the right-angled building introduced in Subsection 2.4. The purpose of this subsection is to identify a natural contractible G G G G = π 1 Z L ( A , B ) G=\pi_1 Z_L(A,B) G = π 1 Z L ( A , B )
Throughout this subsection we first work in the clean situation in which
∣ E i ∣ > 1 for every i ∈ I , |E_i|>1 \qquad\text{for every } i\in I, ∣ E i ∣ > 1 for every i ∈ I , and L L L E i E_i E i
Recall that G i = π 1 ( A ( i ) ) G_i=\pi_1(A(i)) G i = π 1 ( A ( i )) E i = π 0 ( B ~ ( i ) ) E_i=\pi_0(\widetilde{B}(i)) E i = π 0 ( B ( i )) G = Z L ( G , E ) ≅ π 1 Z L ( A , B ) \mathcal{G}=Z_L(G,E)\cong\pi_1Z_L(A,B) G = Z L ( G , E ) ≅ π 1 Z L ( A , B ) i ∈ I i\in I i ∈ I N i = ker ( G i ↷ E i ) N_i=\ker(G_i\curvearrowright E_i) N i = ker ( G i ↷ E i ) N = ∏ i ∈ I N i N=\prod_{i\in I}N_i N = ∏ i ∈ I N i
Let C = ∏ i ∈ I E i C=\prod_{i\in I}E_i C = ∏ i ∈ I E i Z L ( Cone ( E ) , E ) ≅ U ( C , K ( L ) ) Z_L(\operatorname{Cone}(E),E)\cong U(C,K(L)) Z L ( Cone ( E ) , E ) ≅ U ( C , K ( L )) B ~ L = Z L ( Cone ( E ~ ) , E ) \widetilde{\mathcal{B}}_L=Z_L(\operatorname{Cone}(\widetilde{E}),E) B L = Z L ( Cone ( E ) , E ) [8 Proposition 2.10] , there exists a right-angled building C ~ \widetilde{C} C ( W L 1 , S ) (W_{L_1},S) ( W L 1 , S ) B ~ L ≅ U ( C ~ , K ( L ) ) \widetilde{\mathcal{B}}_L\cong U(\widetilde{C},K(L)) B L ≅ U ( C , K ( L )) L L L C ~ \widetilde{C} C B ~ L \widetilde{\mathcal{B}}_L B L
We next describe the action of G \mathcal{G} G Γ L \Gamma_L Γ L ( ∏ i ∈ I G i ) / N (\prod_{i\in I}G_i)/N ( ∏ i ∈ I G i ) / N Z L ( Cone ( E ) , E ) Z_L(\operatorname{Cone}(E),E) Z L ( Cone ( E ) , E )
1 ⟶ π 1 Z L ( Cone ( E ) , E ) ⟶ Γ L ⟶ ( ∏ i ∈ I G i ) / N ⟶ 1. 1\longrightarrow\pi_1Z_L(\operatorname{Cone}(E),E)\longrightarrow\Gamma_L\longrightarrow\left(\prod_{i\in I}G_i\right)/N\longrightarrow1. 1 ⟶ π 1 Z L ( Cone ( E ) , E ) ⟶ Γ L ⟶ ( i ∈ I ∏ G i ) / N ⟶ 1. The relative graph product G \mathcal{G} G ρ : G ⟶ Γ L \rho:\mathcal{G}\longrightarrow\Gamma_L ρ : G ⟶ Γ L G ↷ B ~ L \mathcal{G}\curvearrowright\widetilde{\mathcal{B}}_L G ↷ B L g ⋅ x = ρ ( g ) x g\cdot x=\rho(g)x g ⋅ x = ρ ( g ) x N N N ker ρ = N \ker\rho=N ker ρ = N
The action of Γ L \Gamma_L Γ L G \mathcal{G} G c ∈ C ~ c\in\widetilde{C} c ∈ C J ⊆ I J\subseteq I J ⊆ I Res J ( c ) \operatorname{Res}_J(c) Res J ( c ) J J J c c c J ∈ S ( L ) J\in S(L) J ∈ S ( L ) J J J K ( L ) K(L) K ( L ) J 0 ⊊ J 1 ⊊ ⋯ ⊊ J k J_0\subsetneq J_1\subsetneq\cdots\subsetneq J_k J 0 ⊊ J 1 ⊊ ⋯ ⊊ J k S ( L ) S(L) S ( L ) x x x I ( x ) = J 0 I(x)=J_0 I ( x ) = J 0 U ( C ~ , K ( L ) ) U(\widetilde{C},K(L)) U ( C , K ( L )) [ c , x ] [c,x] [ c , x ] J 0 J_0 J 0 c c c
The standard realization carries its natural cubical structure. Its cubes are indexed by residues of spherical type J ∈ S ( L ) J\in S(L) J ∈ S ( L ) J J J ∣ J ∣ |J| ∣ J ∣
Proposition 3.13. Let σ \sigma σ J ∈ S ( L ) J\in S(L) J ∈ S ( L ) B ~ L \widetilde{\mathcal{B}}_L B L Res J ( c ) \operatorname{Res}_J(c) Res J ( c ) J J J Stab Γ L ( σ ) = Stab Γ L ( Res J ( c ) ) \operatorname{Stab}_{\Gamma_L}(\sigma)=\operatorname{Stab}_{\Gamma_L}(\operatorname{Res}_J(c)) Stab Γ L ( σ ) = Stab Γ L ( Res J ( c )) Stab G ( σ ) = ρ − 1 ( Stab Γ L ( Res J ( c ) ) ) \operatorname{Stab}_{\mathcal{G}}(\sigma)=\rho^{-1}\bigl(\operatorname{Stab}_{\Gamma_L}(\operatorname{Res}_J(c))\bigr) Stab G ( σ ) = ρ − 1 ( Stab Γ L ( Res J ( c )) )
Proof. Let σ \sigma σ J ∈ S ( L ) J\in S(L) J ∈ S ( L ) Res J ( c ) \operatorname{Res}_J(c) Res J ( c ) J J J U ( C ~ , K ( L ) ) U(\widetilde{C},K(L)) U ( C , K ( L )) J J J J J J γ ∈ Γ L \gamma\in\Gamma_L γ ∈ Γ L σ \sigma σ J J J
Indeed, if γ ⋅ σ = σ \gamma\cdot\sigma=\sigma γ ⋅ σ = σ γ \gamma γ σ \sigma σ γ Res J ( c ) = Res J ( c ) \gamma\operatorname{Res}_J(c)=\operatorname{Res}_J(c) γ Res J ( c ) = Res J ( c ) γ Res J ( c ) = Res J ( c ) \gamma\operatorname{Res}_J(c)=\operatorname{Res}_J(c) γ Res J ( c ) = Res J ( c ) γ \gamma γ J J J σ \sigma σ Stab Γ L ( σ ) = Stab Γ L ( Res J ( c ) ) \operatorname{Stab}_{\Gamma_L}(\sigma)=\operatorname{Stab}_{\Gamma_L}(\operatorname{Res}_J(c)) Stab Γ L ( σ ) = Stab Γ L ( Res J ( c ))
The action of G \mathcal{G} G B ~ L \widetilde{\mathcal{B}}_L B L ρ : G ⟶ Γ L \rho:\mathcal{G}\longrightarrow\Gamma_L ρ : G ⟶ Γ L Stab G ( σ ) = ρ − 1 ( Stab Γ L ( σ ) ) \operatorname{Stab}_{\mathcal{G}}(\sigma)=\rho^{-1}(\operatorname{Stab}_{\Gamma_L}(\sigma)) Stab G ( σ ) = ρ − 1 ( Stab Γ L ( σ )) Stab G ( σ ) = ρ − 1 ( Stab Γ L ( Res J ( c ) ) ) \operatorname{Stab}_{\mathcal{G}}(\sigma)=\rho^{-1}(\operatorname{Stab}_{\Gamma_L}(\operatorname{Res}_J(c))) Stab G ( σ ) = ρ − 1 ( Stab Γ L ( Res J ( c )))
Finally, since N = ker ρ N=\ker\rho N = ker ρ N N N B ~ L \widetilde{\mathcal{B}}_L B L N ⊆ Stab G ( σ ) N\subseteq\operatorname{Stab}_{\mathcal{G}}(\sigma) N ⊆ Stab G ( σ ) σ \sigma σ
The preceding description simplifies considerably for ordinary graph products.
Proposition 3.15. Suppose that every G i G_i G i E i E_i E i E i ≅ G i E_i \cong G_i E i ≅ G i G i G_i G i N i = 1 N_i=1 N i = 1 G = ∗ L 1 G i G=*_{L_1}G_i G = ∗ L 1 G i J ∈ S ( L ) J\in S(L) J ∈ S ( L ) J J J G J = ⟨ G j ∣ j ∈ J ⟩ G_J=\langle G_j\mid j\in J\rangle G J = ⟨ G j ∣ j ∈ J ⟩ L L L J J J L 1 L_1 L 1 G J ≅ ∏ j ∈ J G j G_J\cong\prod_{j\in J}G_j G J ≅ ∏ j ∈ J G j Stab G ( σ ) ≅ g ( ∏ j ∈ J G j ) g − 1 \operatorname{Stab}_G(\sigma)\cong g\left(\prod_{j\in J}G_j\right)g^{-1} Stab G ( σ ) ≅ g ( ∏ j ∈ J G j ) g − 1 g ∈ G g\in G g ∈ G
Proof. When the actions G i ↷ E i G_i\curvearrowright E_i G i ↷ E i [8 Remark 2.15] ) agrees with the ordinary graph product. The associated building is the standard right-angled building for this graph product. A cell of type J J J J J J J J J G j G_j G j j ∈ J j\in J j ∈ J J J J J ∈ S ( L ) J\in S(L) J ∈ S ( L ) L L L L 1 L_1 L 1 G J ≅ ∏ j ∈ J G j G_J\cong\prod_{j\in J}G_j G J ≅ ∏ j ∈ J G j
Example 3.16. For ( A ( i ) , B ( i ) ) = ( B G i , ∗ ) (A(i),B(i))=(BG_i,*) ( A ( i ) , B ( i )) = ( B G i , ∗ ) E i E_i E i G i G_i G i π 1 Z L 1 ( B G , ∗ ) ≅ ∗ L 1 G i \pi_1Z_{L_1}(BG,*)\cong*_{L_1}G_i π 1 Z L 1 ( B G , ∗ ) ≅ ∗ L 1 G i J J J ∏ j ∈ J G j \prod_{j\in J}G_j ∏ j ∈ J G j
We now give a more explicit description in the transitive case, using the graph products of pairs introduced by Januszkiewicz and Świątkowski in [JŚ01].
Suppose that each G i G_i G i E i E_i E i e i ∈ E i e_i\in E_i e i ∈ E i H i = Stab G i ( e i ) H_i=\operatorname{Stab}_{G_i}(e_i) H i = Stab G i ( e i ) E i ≅ G i / H i E_i\cong G_i/H_i E i ≅ G i / H i Z L 1 ( G , E ) Z_{L_1}(G,E) Z L 1 ( G , E ) Π L 1 ( G i , H i ) \Pi_{L_1}(G_i,H_i) Π L 1 ( G i , H i ) J ∈ S ( L ) J\in S(L) J ∈ S ( L )
G J = ∏ j ∈ J G j × ∏ i ∈ I ∖ J H i . G_J=\prod_{j\in J}G_j\times\prod_{i\in I\setminus J}H_i. G J = j ∈ J ∏ G j × i ∈ I ∖ J ∏ H i . These groups form the local groups of the simple complex of groups associated with the graph product of pairs. Januszkiewicz and Świątkowski show that the corresponding development is the right-angled building associated with the graph product, and that the complex of groups arising from the group action on this building has precisely the local groups G J G_J G J
Proposition 3.17. Assume that every action G i ↷ E i G_i\curvearrowright E_i G i ↷ E i H i = Stab G i ( e i ) H_i=\operatorname{Stab}_{G_i}(e_i) H i = Stab G i ( e i ) e i ∈ E i e_i\in E_i e i ∈ E i G = Z L 1 ( G , E ) ≅ Π L 1 ( G i , H i ) G=Z_{L_1}(G,E)\cong\Pi_{L_1}(G_i,H_i) G = Z L 1 ( G , E ) ≅ Π L 1 ( G i , H i ) J ∈ S ( L ) J\in S(L) J ∈ S ( L ) J J J
G J = ∏ j ∈ J G j × ∏ i ∈ I ∖ J H i . G_J=\prod_{j\in J}G_j\times\prod_{i\in I\setminus J}H_i. G J = j ∈ J ∏ G j × i ∈ I ∖ J ∏ H i . Proof. Since the actions G i ↷ E i G_i\curvearrowright E_i G i ↷ E i E i ≅ G i / H i E_i\cong G_i/H_i E i ≅ G i / H i [8 Remark 2.15] , M. Davis identifies the corresponding relative graph product with the graph product of pairs Π L 1 ( G i , H i ) \Pi_{L_1}(G_i,H_i) Π L 1 ( G i , H i )
The graph product of pairs is the direct limit of the system of local groups
G J = ∏ j ∈ J G j × ∏ i ∈ I ∖ J H i , J ∈ S ( L ) . G_J=\prod_{j\in J}G_j\times\prod_{i\in I\setminus J}H_i,\qquad J\in S(L). G J = j ∈ J ∏ G j × i ∈ I ∖ J ∏ H i , J ∈ S ( L ) . Januszkiewicz and Świątkowski construct the associated complex D G D_G D G G G G D G D_G D G G J G_J G J J J J G J G_J G J J J J G J G_J G J
Remark 3.18. For J = ∅ J=\varnothing J = ∅ G ∅ = ∏ i ∈ I H i G_{\varnothing}=\prod_{i\in I}H_i G ∅ = ∏ i ∈ I H i
On the other hand, if H i = { 1 } H_i=\{1\} H i = { 1 } i i i G J = ∏ j ∈ J G j G_J=\prod_{j\in J}G_j G J = ∏ j ∈ J G j Proposition 3.17 reduces to Proposition 3.15 .
Remark 3.19. Let N i = ⋂ g ∈ G i g H i g − 1 N_i=\bigcap_{g\in G_i}gH_ig^{-1} N i = ⋂ g ∈ G i g H i g − 1 H i H_i H i G i G_i G i N i N_i N i G i ↷ G i / H i G_i\curvearrowright G_i/H_i G i ↷ G i / H i N = ∏ i ∈ I N i N=\prod_{i\in I}N_i N = ∏ i ∈ I N i G J G_J G J Proposition 3.13 , according to which the kernel of the action is contained in every cell stabilizer.
For arbitrary, not necessarily transitive, G i G_i G i E i E_i E i J J J J J J Proposition 3.13 . Thus the relevant local groups for the sequel are G c , J : = ρ − 1 ( Stab Γ L ( Res J ( c ) ) ) G_{c,J}:=\rho^{-1}(\operatorname{Stab}_{\Gamma_L}(\operatorname{Res}_J(c))) G c , J := ρ − 1 ( Stab Γ L ( Res J ( c ))) Stab G ( σ ) = G c , J \operatorname{Stab}_{G}(\sigma)=G_{c,J} Stab G ( σ ) = G c , J σ \sigma σ J J J c c c
Although G c , J G_{c,J} G c , J G G G G G G { G c , J } \{G_{c,J}\} { G c , J }
Finally, suppose that some of the sets E i E_i E i
I ′ ′ = { i ∈ I ∣ ∣ E i ∣ > 1 } , L ′ ′ = L I ′ ′ . I''=\{i\in I\mid|E_i|>1\},\qquad L''=L_{I''}. I ′′ = { i ∈ I ∣ ∣ E i ∣ > 1 } , L ′′ = L I ′′ . The nontrivial right-angled building is then associated with L ′ ′ L'' L ′′ L L L L ′ ′ L'' L ′′ I ′ ′ I'' I ′′
We have therefore obtained a contractible G G G B ~ L \widetilde{B}_L B L
Combination theorems for polyhedral-product groups
Families of subgroups and F \mathcal{F} F
We begin by recalling the notion of F \mathcal{F} F
Let G G G family of subgroups of G G G F \mathcal{F} F G G G
Tr , Fin , Am , \operatorname{Tr},\qquad\operatorname{Fin},\qquad\operatorname{Am}, Tr , Fin , Am , the families consisting, respectively, of the trivial subgroup, the finite subgroups, and the amenable subgroups of G G G
Let Y Y Y G = π 1 ( Y ) G=\pi_1(Y) G = π 1 ( Y ) [7 Definition 1.2] , the F \mathcal{F} F Y Y Y cat F ( Y ) \operatorname{cat}_{\mathcal{F}}(Y) cat F ( Y ) n ∈ N n\in\mathbb{N} n ∈ N Y = U 0 ∪ ⋯ ∪ U n Y=U_0\cup\cdots\cup U_n Y = U 0 ∪ ⋯ ∪ U n i ∈ { 0 , … , n } i\in\{0,\ldots,n\} i ∈ { 0 , … , n } y ∈ U i y\in U_i y ∈ U i π 1 ( U i , y ) ⟶ π 1 ( Y , y ) \pi_1(U_i,y)\longrightarrow\pi_1(Y,y) π 1 ( U i , y ) ⟶ π 1 ( Y , y ) F \mathcal{F} F
For a group G G G cat F ( G ) : = cat F ( B G ) \operatorname{cat}_{\mathcal{F}}(G):=\operatorname{cat}_{\mathcal{F}}(BG) cat F ( G ) := cat F ( B G ) B G BG B G Y ⟶ Z Y\longrightarrow Z Y ⟶ Z cat F ( Y ) ≤ cat F ( Z ) \operatorname{cat}_{\mathcal{F}}(Y)\leq\operatorname{cat}_{\mathcal{F}}(Z) cat F ( Y ) ≤ cat F ( Z ) cat F ( Y ) ≤ cat F ( π 1 ( Y ) ) \operatorname{cat}_{\mathcal{F}}(Y)\leq\operatorname{cat}_{\mathcal{F}}(\pi_1(Y)) cat F ( Y ) ≤ cat F ( π 1 ( Y ))
If E ⊆ F \mathcal{E}\subseteq\mathcal{F} E ⊆ F G G G cat F ( Y ) ≤ cat E ( Y ) \operatorname{cat}_{\mathcal{F}}(Y)\leq\operatorname{cat}_{\mathcal{E}}(Y) cat F ( Y ) ≤ cat E ( Y )
cat F ( G ) = 0 ⟺ G ∈ F . \operatorname{cat}_{\mathcal{F}}(G)=0\quad\Longleftrightarrow\quad G\in\mathcal{F}. cat F ( G ) = 0 ⟺ G ∈ F . For later use, if H ≤ G H\leq G H ≤ G F ∣ H = { F ′ ≤ H ∣ F ′ ∈ F } \mathcal{F}_{|H}=\{F'\leq H\mid F'\in\mathcal{F}\} F ∣ H = { F ′ ≤ H ∣ F ′ ∈ F } F \mathcal{F} F H H H
The F \mathcal{F} F X X X G G G cat F G ( X ) \operatorname{cat}_{\mathcal{F}}^G(X) cat F G ( X ) n ∈ N n\in\mathbb{N} n ∈ N n n n G G G N N N F \mathcal{F} F G G G X ⟶ N X \longrightarrow N X ⟶ N with isotropy in F \mathcal{F} F means that the stabilizer of every cell of N N N F \mathcal{F} F
The equivariant invariant is monotone under G G G X 1 ⟶ X 2 X_1 \longrightarrow X_2 X 1 ⟶ X 2 G G G cat F G ( X 1 ) ≤ cat F G ( X 2 ) \operatorname{cat}_{\mathcal{F}}^G(X_1) \le\operatorname{cat}_{\mathcal{F}}^G(X_2) cat F G ( X 1 ) ≤ cat F G ( X 2 ) cat F G \operatorname{cat}_{\mathcal{F}}^G cat F G G G G
The connection with the nonequivariant definition is the following.
Proposition 4.1 (Capovilla–Li–Löh cf. [7 ] ). Let Y Y Y G = π 1 ( Y ) G=\pi_1(Y) G = π 1 ( Y ) Y ~ \widetilde{Y} Y G G G cat F ( Y ) = cat F G ( Y ~ ) \operatorname{cat}_{\mathcal{F}}(Y)=\operatorname{cat}_{\mathcal{F}}^G(\widetilde{Y}) cat F ( Y ) = cat F G ( Y ) cat F ( G ) = cat F G ( E G ) \operatorname{cat}_{\mathcal{F}}(G)=\operatorname{cat}_{\mathcal{F}}^G(EG) cat F ( G ) = cat F G ( E G )
This reformulation is the key point for our purposes. Indeed, if X X X G G G X X X F \mathcal{F} F G G G B ~ L \widetilde{B}_L B L G G G
We record several special cases. For the trivial family Tr = { { 1 } } \operatorname{Tr}=\{\{1\}\} Tr = {{ 1 }} cat Tr ( G ) = cd ( G ) \operatorname{cat}_{\operatorname{Tr}}(G)=\operatorname{cd}(G) cat Tr ( G ) = cd ( G ) Fin \operatorname{Fin} Fin Am \operatorname{Am} Am cat Am ( G ) \operatorname{cat}_{\operatorname{Am}}(G) cat Am ( G )
The usefulness of the equivariant formulation is that it permits one to estimate global invariants of G G G G G G G = π 1 Z L ( A , B ) G=\pi_1 Z_L(A,B) G = π 1 Z L ( A , B )
Combination via contractible G G G
We now recall the combination theorem of Capovilla–Li–Löh [7 ] , which provides the group-theoretic input for our applications to polyhedral products.
Let G G G F \mathcal{F} F G G G X X X G G G σ ⊆ X \sigma\subseteq X σ ⊆ X G σ = Stab G ( σ ) G_\sigma=\operatorname{Stab}_G(\sigma) G σ = Stab G ( σ ) i ≥ 0 i\ge0 i ≥ 0 Σ i \Sigma_i Σ i G G G i i i X X X
The basic principle is that, when X X X F \mathcal{F} F G G G gd ( G ) \operatorname{gd}(G) gd ( G ) B G BG B G G G G gd ( G σ ) \operatorname{gd}(G_\sigma) gd ( G σ ) cat F ∣ G σ ( G σ ) \operatorname{cat}_{\mathcal{F}|_{G_\sigma}}(G_\sigma) cat F ∣ G σ ( G σ )
Theorem 4.2 (Capovilla–Li–Löh). Let G G G F \mathcal{F} F G G G X X X n n n G G G i ∈ { 0 , … , n } i\in\{0,\ldots,n\} i ∈ { 0 , … , n } Σ i \Sigma_i Σ i G G G i i i X X X
Fix a subset I ⊆ { 1 , … , n } \mathcal{I}\subseteq\{1,\ldots,n\} I ⊆ { 1 , … , n } d 0 = sup ν ∈ Σ 0 cat F ∣ G ν ( G ν ) d_0=\sup_{\nu\in\Sigma_0}\operatorname{cat}_{\mathcal{F}|_{G_\nu}}(G_\nu) d 0 = sup ν ∈ Σ 0 cat F ∣ G ν ( G ν ) 1 ≤ i ≤ n 1\le i\le n 1 ≤ i ≤ n
d i = { max { d i − 1 , sup σ ∈ Σ i ( gd ( G σ ) + i ) } , i ∈ I , d i − 1 + sup σ ∈ Σ i ( cat F ∣ G σ ( G σ ) + 1 ) , i ∉ I . d_i=
\begin{cases}
\max\left\{d_{i-1},\sup_{\sigma\in\Sigma_i}\left(\operatorname{gd}(G_\sigma)+i\right)\right\}, & i\in\mathcal{I},\\
d_{i-1}+\sup_{\sigma\in\Sigma_i}\left(\operatorname{cat}_{\mathcal{F}|_{G_\sigma}}(G_\sigma)+1\right), & i\notin\mathcal{I}.
\end{cases} d i = { max { d i − 1 , sup σ ∈ Σ i ( gd ( G σ ) + i ) } , d i − 1 + sup σ ∈ Σ i ( cat F ∣ G σ ( G σ ) + 1 ) , i ∈ I , i ∈ / I . Then cat F G ( E G × X ) ≤ d n \operatorname{cat}_{\mathcal{F}}^G(EG\times X)\le d_n cat F G ( E G × X ) ≤ d n G G G E G × X EG\times X E G × X
If, in addition, X X X cat F ( G ) ≤ d n \operatorname{cat}_{\mathcal{F}}(G)\le d_n cat F ( G ) ≤ d n
The strength of Theorem 4.2 lies in the freedom to choose I \mathcal{I} I F \mathcal{F} F
We record the two extreme choices separately.
Corollary 4.3. Let G G G G G G X X X Σ 0 \Sigma_0 Σ 0 G G G X X X Σ ≥ 1 \Sigma_{\geq1} Σ ≥ 1 G G G
cat F ( G ) ≤ max { sup v ∈ Σ 0 cat F ∣ G v ( G v ) , sup σ ∈ Σ ≥ 1 ( gd ( G σ ) + dim σ ) } . \operatorname{cat}_{\mathcal{F}}(G) \leq\max\left\{ \sup_{v \in\Sigma_0} \operatorname{cat}_{\mathcal{F}|G_v}(G_v), \sup_{\sigma\in\Sigma_{\geq1}} \left(\operatorname{gd}(G_\sigma) + \dim\sigma\right) \right\}. cat F ( G ) ≤ max { v ∈ Σ 0 sup cat F ∣ G v ( G v ) , σ ∈ Σ ≥ 1 sup ( gd ( G σ ) + dim σ ) } . Proof. Apply Theorem 4.2 with I = { 1 , … , dim X } \mathcal{I} = \{1,\ldots,\dim X\} I = { 1 , … , dim X }
At the opposite extreme one obtains an estimate involving only F \mathcal{F} F
Corollary 4.4. Let G G G n n n G G G X X X
cat F ( G ) ≤ sup v ∈ Σ 0 cat F ∣ G v ( G v ) + ∑ i = 1 n sup σ ∈ Σ i ( cat F ∣ G σ ( G σ ) + 1 ) . \operatorname{cat}_{\mathcal{F}}(G) \leq\sup_{v \in\Sigma_0} \operatorname{cat}_{\mathcal{F}|G_v}(G_v) + \sum_{i=1}^{n} \sup_{\sigma\in\Sigma_i} \left(\operatorname{cat}_{\mathcal{F}|G_\sigma}(G_\sigma) + 1\right). cat F ( G ) ≤ v ∈ Σ 0 sup cat F ∣ G v ( G v ) + i = 1 ∑ n σ ∈ Σ i sup ( cat F ∣ G σ ( G σ ) + 1 ) . Proof. Apply Theorem 4.2 with I = ∅ \mathcal{I} = \varnothing I = ∅
Remark 4.5. Neither of the preceding estimates dominates the other in general. The geometric-dimension bound is particularly effective when the positive-dimensional cell stabilizers have small geometric dimension, whereas the second estimate may be sharper when their F \mathcal{F} F Theorem 4.2 allows these two phenomena to be combined.
The relevance of these results to polyhedral products is now immediate at the level of principle. In Subsection 3.3, we associated to an aspherical polyhedral product a contractible right-angled building equipped with an action of its fundamental group and described the cell stabilizers in terms of residue stabilizers. We now insert this geometric model into Theorem 4.2 .
The main combination theorem
We now apply the preceding combination principle to fundamental groups of aspherical polyhedral products.
Let Z = Z L ( A , B ) Z = Z_L(A,B) Z = Z L ( A , B ) ∣ E i ∣ > 1 \lvert E_i\rvert> 1 ∣ E i ∣ > 1 i ∈ I i \in I i ∈ I L L L G = π 1 ( Z ) ≅ Z L 1 ( G , E ) G = \pi_1(Z) \cong Z_{L_1}(G,E) G = π 1 ( Z ) ≅ Z L 1 ( G , E ) G i = π 1 ( A ( i ) ) G_i = \pi_1(A(i)) G i = π 1 ( A ( i )) E i = π 0 ( B ~ ( i ) ) E_i = \pi_0(\widetilde{B}(i)) E i = π 0 ( B ( i ))
By Subsection 3.3, G G G B L = U ( C ~ , K ( L ) ) B_L = \mathcal{U}(\widetilde{C},K(L)) B L = U ( C , K ( L )) L L L B L B_L B L
Let n = dim B L = dim L + 1 n = \dim B_L = \dim L + 1 n = dim B L = dim L + 1 i ∈ { 0 , … , n } i \in\{0,\ldots,n\} i ∈ { 0 , … , n } Σ i \Sigma_i Σ i G G G i i i B L B_L B L
If σ ∈ Σ i \sigma\in\Sigma_i σ ∈ Σ i σ \sigma σ J ( σ ) ∈ S ( L ) J(\sigma) \in S(L) J ( σ ) ∈ S ( L ) ∣ J ( σ ) ∣ = i \lvert J(\sigma)\rvert= i ∣ J ( σ )∣ = i c σ c_\sigma c σ σ \sigma σ H σ : = G c σ , J ( σ ) = Stab G ( σ ) H_\sigma:= G_{c_\sigma,J(\sigma)} = \operatorname{Stab}_G(\sigma) H σ := G c σ , J ( σ ) = Stab G ( σ )
H σ = ρ − 1 ( Stab Γ L ( Res J ( σ ) ( c σ ) ) ) . H_\sigma= \rho^{-1}\left(\operatorname{Stab}_{\Gamma_L}\left(\operatorname{Res}_{J(\sigma)}(c_\sigma)\right)\right). H σ = ρ − 1 ( Stab Γ L ( Res J ( σ ) ( c σ ) ) ) . We can now state the main combination theorem.
Theorem 4.6. Let Z = Z L ( A , B ) Z = Z_L(A,B) Z = Z L ( A , B ) G = π 1 ( Z ) G = \pi_1(Z) G = π 1 ( Z ) F \mathcal{F} F G G G I ⊆ { 1 , … , n } \mathcal{I} \subseteq\{1,\ldots,n\} I ⊆ { 1 , … , n } n = dim L + 1 n = \dim L + 1 n = dim L + 1
d 0 = sup σ ∈ Σ 0 cat F ∣ H σ ( H σ ) , d_0 = \sup_{\sigma\in\Sigma_0} \operatorname{cat}_{\mathcal{F}|H_\sigma}(H_\sigma), d 0 = σ ∈ Σ 0 sup cat F ∣ H σ ( H σ ) , and, for 1 ≤ i ≤ n 1 \leq i \leq n 1 ≤ i ≤ n
d i = { max { d i − 1 , sup σ ∈ Σ i ( gd ( H σ ) + i ) } , i ∈ I , d i − 1 + sup σ ∈ Σ i ( cat F ∣ H σ ( H σ ) + 1 ) , i ∉ I . d_i =
\begin{cases}
\max\left\{d_{i-1}, \sup_{\sigma\in\Sigma_i}\left(\operatorname{gd}(H_\sigma)+i\right)\right\}, & i \in\mathcal{I},\\
d_{i-1}+\sup_{\sigma\in\Sigma_i}\left(\operatorname{cat}_{\mathcal{F}|H_\sigma}(H_\sigma)+1\right), & i \notin\mathcal{I}.
\end{cases} d i = { max { d i − 1 , sup σ ∈ Σ i ( gd ( H σ ) + i ) } , d i − 1 + sup σ ∈ Σ i ( cat F ∣ H σ ( H σ ) + 1 ) , i ∈ I , i ∈ / I . Then cat F ( π 1 Z L ( A , B ) ) ≤ d n \operatorname{cat}_{\mathcal{F}}(\pi_{1}Z_{L}(A,B)) \le d_{n} cat F ( π 1 Z L ( A , B )) ≤ d n
Proof. By Subsection 3.3, the group G = π 1 Z L ( A , B ) G=\pi_{1}Z_{L}(A,B) G = π 1 Z L ( A , B ) B L = U ( C ~ , K ( L ) ) \mathcal{B}_{L}=\mathcal{U}(\widetilde{C},K(L)) B L = U ( C , K ( L )) L L L B L \mathcal{B}_{L} B L σ ⊆ B L \sigma\subseteq\mathcal{B}_{L} σ ⊆ B L G σ = H σ = G c σ , J ( σ ) G_{\sigma}=H_{\sigma}=\mathcal{G}_{c_{\sigma},J(\sigma)} G σ = H σ = G c σ , J ( σ ) Proposition 3.13 . Theorem 4.2 applied to the contractible G G G B L \mathcal{B}_{L} B L cat F ( G ) ≤ d n \operatorname{cat}_{\mathcal{F}}(G)\le d_{n} cat F ( G ) ≤ d n G ≅ π 1 Z L ( A , B ) G\cong\pi_{1}Z_{L}(A,B) G ≅ π 1 Z L ( A , B )
The two extreme choices of I \mathcal{I} I Theorem 4.6 .
Corollary 4.7. Under the assumptions of Theorem 4.6 , one has
cat F ( G ) ≤ max { sup σ ∈ Σ 0 cat F ∣ H σ ( H σ ) , sup σ ⊆ B L dim σ > 0 ( gd ( H σ ) + dim σ ) } . \operatorname{cat}_{\mathcal{F}}(G)\le\max\left\{\sup_{\sigma\in\Sigma_{0}}\operatorname{cat}_{\mathcal{F}|H_{\sigma}}(H_{\sigma}),\ \sup_{\substack{\sigma\subseteq\mathcal{B}_{L}\\ \dim\sigma>0}}\left(\operatorname{gd}(H_{\sigma})+\dim\sigma\right)\right\}. cat F ( G ) ≤ max ⎩ ⎨ ⎧ σ ∈ Σ 0 sup cat F ∣ H σ ( H σ ) , σ ⊆ B L d i m σ > 0 sup ( gd ( H σ ) + dim σ ) ⎭ ⎬ ⎫ . Proof. Take I = { 1 , … , n } \mathcal{I}=\{1,\ldots,n\} I = { 1 , … , n } Theorem 4.6 . □
Corollary 4.8. Under the same assumptions,
cat F ( G ) ≤ sup σ ∈ Σ 0 cat F ∣ H σ ( H σ ) + ∑ i = 1 n sup σ ∈ Σ i ( cat F ∣ H σ ( H σ ) + 1 ) . \operatorname{cat}_{\mathcal{F}}(G)\le\sup_{\sigma\in\Sigma_{0}}\operatorname{cat}_{\mathcal{F}|H_{\sigma}}(H_{\sigma})+\sum_{i=1}^{n}\sup_{\sigma\in\Sigma_{i}}\left(\operatorname{cat}_{\mathcal{F}|H_{\sigma}}(H_{\sigma})+1\right). cat F ( G ) ≤ σ ∈ Σ 0 sup cat F ∣ H σ ( H σ ) + i = 1 ∑ n σ ∈ Σ i sup ( cat F ∣ H σ ( H σ ) + 1 ) . Proof. Take I = ∅ \mathcal{I}=\varnothing I = ∅ Theorem 4.6 .
Graph products of pairs
We now specialize the preceding combination theorem to the transitive relative case. Besides producing explicit bounds, this connects the polyhedral-product construction with the graph products of pairs of Januszkiewicz and Świątkowski [JŚ01].
Assume that E i ≅ G i / H i E_{i}\cong G_{i}/H_{i} E i ≅ G i / H i i ∈ I i\in I i ∈ I G = Π L 1 ( G i , H i ) ≅ Z L ( G , E ) G=\Pi_{L}^{1}(G_{i},H_{i})\cong Z_{L}(G,E) G = Π L 1 ( G i , H i ) ≅ Z L ( G , E ) J ∈ S ( L ) J\in S(L) J ∈ S ( L ) G J = ∏ j ∈ J G j × ∏ i ∈ I ∖ J H i G_{J}=\prod_{j\in J}G_{j}\times\prod_{i\in I\setminus J}H_{i} G J = ∏ j ∈ J G j × ∏ i ∈ I ∖ J H i Proposition 3.17 , the stabilizer of a cube of type J J J G J G_{J} G J
We therefore obtain the following explicit form of the combination estimate.
Theorem 4.9. Let L L L G = Π L 1 ( G i , H i ) G=\Pi_{L}^{1}(G_{i},H_{i}) G = Π L 1 ( G i , H i ) F \mathcal{F} F G G G
cat F ( G ) ≤ max { cat F ∣ G ∅ ( G ∅ ) , max ∅ ≠ J ∈ S ( L ) ( gd ( G J ) + ∣ J ∣ ) } . \operatorname{cat}_{\mathcal{F}}(G)\le\max\left\{\operatorname{cat}_{\mathcal{F}|G_{\varnothing}}(G_{\varnothing}),\ \max_{\varnothing\ne J\in S(L)}\left(\operatorname{gd}(G_{J})+|J|\right)\right\}. cat F ( G ) ≤ max { cat F ∣ G ∅ ( G ∅ ) , ∅ = J ∈ S ( L ) max ( gd ( G J ) + ∣ J ∣ ) } . where G ∅ = ∏ i ∈ I H i G_{\varnothing}=\prod_{i\in I}H_{i} G ∅ = ∏ i ∈ I H i G J = ∏ j ∈ J G j × ∏ i ∉ J H i G_{J}=\prod_{j\in J}G_{j}\times\prod_{i\notin J}H_{i} G J = ∏ j ∈ J G j × ∏ i ∈ / J H i
Proof. Let D G D_{G} D G L L L Proposition 3.17 , a cube of type J ∈ S ( L ) J\in S(L) J ∈ S ( L ) G J G_{J} G J J J J ∣ J ∣ |J| ∣ J ∣ G ∅ = ∏ i ∈ I H i G_{\varnothing}=\prod_{i\in I}H_{i} G ∅ = ∏ i ∈ I H i G G G D G D_{G} D G
cat F ( G ) ≤ max { cat F ∣ G ∅ ( G ∅ ) , max ∅ ≠ J ∈ S ( L ) ( gd ( G J ) + ∣ J ∣ ) } . \operatorname{cat}_{\mathcal{F}}(G)\le\max\left\{\operatorname{cat}_{\mathcal{F}|G_{\varnothing}}(G_{\varnothing}),\ \max_{\varnothing\ne J\in S(L)}\left(\operatorname{gd}(G_{J})+|J|\right)\right\}. cat F ( G ) ≤ max { cat F ∣ G ∅ ( G ∅ ) , ∅ = J ∈ S ( L ) max ( gd ( G J ) + ∣ J ∣ ) } . Corollary 4.10. Suppose that H i = { 1 } H_{i}=\{1\} H i = { 1 } i ∈ I i\in I i ∈ I G = ∗ L 1 G i G=*_{L}^{1}G_{i} G = ∗ L 1 G i G J = ∏ j ∈ J G j G_{J}=\prod_{j\in J}G_{j} G J = ∏ j ∈ J G j
cat F ( G ) ≤ max ∅ ≠ J ∈ S ( L ) { gd ( ∏ j ∈ J G j ) + ∣ J ∣ } . \operatorname{cat}_{\mathcal{F}}(G)\le\max_{\varnothing\ne J\in S(L)}\left\{\operatorname{gd}\left(\prod_{j\in J}G_{j}\right)+|J|\right\}. cat F ( G ) ≤ ∅ = J ∈ S ( L ) max ⎩ ⎨ ⎧ gd j ∈ J ∏ G j + ∣ J ∣ ⎭ ⎬ ⎫ . Proof. This is the special case H i = { 1 } H_{i}=\{1\} H i = { 1 } G ∅ G_{\varnothing} G ∅ cat F ( G ∅ ) = 0 \operatorname{cat}_{\mathcal{F}}(G_{\varnothing})=0 cat F ( G ∅ ) = 0
Applications
We now record several consequences of the combination estimates obtained in the previous section. We concentrate on three invariants for which the F \mathcal{F} F
Amenable category
Let Am \operatorname{Am} Am G G G cat Am ( G ) \operatorname{cat}_{\operatorname{Am}}(G) cat Am ( G ) B G BG B G G G G
Amenable category is particularly useful because it is closely related to bounded cohomology and simplicial volume. We first specialize the combination estimates of the previous section to the family Am \operatorname{Am} Am
Corollary 5.1. Let Z = Z L ( A , B ) Z = Z_L(A,B) Z = Z L ( A , B ) Theorem 4.6 , and put G = π 1 ( Z ) G = \pi_1(Z) G = π 1 ( Z )
cat Am ( G ) ≤ max { sup σ ∈ Σ 0 cat Am ∣ H σ ( H σ ) , sup σ ⊆ B L dim σ > 0 ( gd ( H σ ) + dim σ ) } . \operatorname{cat}_{\operatorname{Am}}(G) \le\max\left\{ \sup_{\sigma\in\Sigma_0} \operatorname{cat}_{\operatorname{Am}\mid H_\sigma}(H_\sigma),\ \sup_{\substack{\sigma\subseteq B_L \\ \dim\sigma> 0}} \left(\operatorname{gd}(H_\sigma) + \dim\sigma\right) \right\}. cat Am ( G ) ≤ max ⎩ ⎨ ⎧ σ ∈ Σ 0 sup cat Am ∣ H σ ( H σ ) , σ ⊆ B L d i m σ > 0 sup ( gd ( H σ ) + dim σ ) ⎭ ⎬ ⎫ . where b H σ = Stab G ( σ ) \mathrm{b}H_\sigma= \operatorname{Stab}_G(\sigma) b H σ = Stab G ( σ )
Proof. This is Corollary 4.7 applied to the family F = Am \mathcal{F} = \operatorname{Am} F = Am
In the transitive case, the stabilizers of the standard right-angled building are explicit, and the preceding estimate can be written entirely in terms of the defining pairs.
Corollary 5.2. Let L L L G = Π L 1 ( G i , H i ) G = \Pi_{L^1}(G_i,H_i) G = Π L 1 ( G i , H i ) J ∈ S ( L ) J \in S(L) J ∈ S ( L ) G J = ∏ j ∈ J G j × ∏ i ∉ J H i G_J = \prod_{j \in J} G_j \times\prod_{i \notin J} H_i G J = ∏ j ∈ J G j × ∏ i ∈ / J H i
cat Am ( G ) ≤ max { cat Am ( ∏ i ∈ I H i ) , max ∅ ≠ J ∈ S ( L ) ( gd ( G J ) + ∣ J ∣ ) } . \operatorname{cat}_{\operatorname{Am}}(G) \le\max\left\{ \operatorname{cat}_{\operatorname{Am}}\left(\prod_{i \in I} H_i\right),\ \max_{\varnothing\ne J \in S(L)} \left(\operatorname{gd}(G_J) + |J|\right) \right\}. cat Am ( G ) ≤ max { cat Am ( i ∈ I ∏ H i ) , ∅ = J ∈ S ( L ) max ( gd ( G J ) + ∣ J ∣ ) } . Proof. By Proposition 3.17 , the stabilizer of a 0-cube is conjugate to G ∅ = ∏ i ∈ I H i G_{\varnothing} = \prod_{i \in I} H_i G ∅ = ∏ i ∈ I H i J ∈ S ( L ) J \in S(L) J ∈ S ( L ) ∣ J ∣ |J| ∣ J ∣ G J G_J G J F = Am \mathcal{F} = \operatorname{Am} F = Am
The estimate simplifies when the base stabilizer is amenable.
Corollary 5.3. Under the hypotheses of Corollary 5.2 , suppose that every H i H_i H i
cat Am ( G ) ≤ max ∅ ≠ J ∈ S ( L ) ( gd ( ∏ j ∈ J G j × ∏ i ∉ J H i ) + ∣ J ∣ ) . \operatorname{cat}_{\operatorname{Am}}(G) \le\max_{\varnothing\ne J \in S(L)} \left( \operatorname{gd}\left(\prod_{j \in J} G_j \times\prod_{i \notin J} H_i\right) + |J| \right). cat Am ( G ) ≤ ∅ = J ∈ S ( L ) max gd j ∈ J ∏ G j × i ∈ / J ∏ H i + ∣ J ∣ . Proof. Finite products of amenable groups are amenable. Hence G ∅ = ∏ i ∈ I H i G_{\varnothing} = \prod_{i \in I} H_i G ∅ = ∏ i ∈ I H i cat Am ( G ∅ ) = 0 \operatorname{cat}_{\operatorname{Am}}(G_{\varnothing}) = 0 cat Am ( G ∅ ) = 0 Corollary 5.2 . □
There is a different estimate when all local groups are amenable. Unlike the preceding bounds, this estimate uses the additive form of the combination theorem.
Corollary 5.4. Let L L L G = Π L 1 ( G i , H i ) G = \Pi_{L^1}(G_i,H_i) G = Π L 1 ( G i , H i ) G i G_i G i cat Am ( G ) ≤ dim L + 1 \operatorname{cat}_{\operatorname{Am}}(G) \le\dim L + 1 cat Am ( G ) ≤ dim L + 1
Proof. Since H i ≤ G i H_i \le G_i H i ≤ G i H i H_i H i J ∈ S ( L ) J \in S(L) J ∈ S ( L ) G J = ∏ j ∈ J G j × ∏ i ∉ J H i G_J = \prod_{j \in J} G_j \times\prod_{i \notin J} H_i G J = ∏ j ∈ J G j × ∏ i ∈ / J H i cat Am ∣ G J ( G J ) = 0 \operatorname{cat}_{\operatorname{Am}\mid G_J}(G_J) = 0 cat Am ∣ G J ( G J ) = 0
cat A m ( G ) ≤ ∑ r = 1 dim L + 1 1 = dim L + 1. \operatorname{cat}_{\mathrm{Am}}(G) \le\sum_{r=1}^{\dim L+1} 1 = \dim L+1. cat Am ( G ) ≤ r = 1 ∑ d i m L + 1 1 = dim L + 1. Remark 5.5. Even when all the vertex groups G i G_i G i G G G
The preceding results require no normality or splitting assumptions on the pairs ( G i , H i ) (G_i,H_i) ( G i , H i )
Amenable extensions and exact computations
The estimates of the previous subsection apply to arbitrary graph products of pairs, but in general they provide only upper bounds. For normal pairs, the graph product of pairs admits a natural quotient which is an ordinary graph product. When the kernels are amenable, this quotient can be used to obtain substantially sharper estimates, and in the split case it leads to exact computations.
We begin with a general observation about amenable extensions.
Lemma 5.6. Let
1 ⟶ N ⟶ G → ρ Q ⟶ 1 1 \longrightarrow N \longrightarrow G \xrightarrow{\rho} Q \longrightarrow1 1 ⟶ N ⟶ G ρ Q ⟶ 1 be a short exact sequence of groups. If N N N cat A m ( G ) ≤ cat A m ( Q ) \operatorname{cat}_{\mathrm{Am}}(G) \le\operatorname{cat}_{\mathrm{Am}}(Q) cat Am ( G ) ≤ cat Am ( Q )
Proof. Let n = cat A m ( Q ) n=\operatorname{cat}_{\mathrm{Am}}(Q) n = cat Am ( Q ) B Q = U 0 ∪ ⋯ ∪ U n BQ=U_0\cup\cdots\cup U_n B Q = U 0 ∪ ⋯ ∪ U n f : B G ⟶ B Q f\colon BG\longrightarrow BQ f : B G ⟶ B Q ρ \rho ρ V i = f − 1 ( U i ) V_i=f^{-1}(U_i) V i = f − 1 ( U i ) V 0 , … , V n V_0,\ldots,V_n V 0 , … , V n B G BG B G
Let V V V V i V_i V i x ∈ V x\in V x ∈ V Γ = im ( π 1 ( V , x ) ⟶ G ) \Gamma=\operatorname{im}(\pi_1(V,x)\longrightarrow G) Γ = im ( π 1 ( V , x ) ⟶ G ) U i U_i U i ρ ( Γ ) ≤ Q \rho(\Gamma)\le Q ρ ( Γ ) ≤ Q Γ ∩ N \Gamma\cap N Γ ∩ N N N N
1 ⟶ Γ ∩ N ⟶ Γ ⟶ ρ ( Γ ) ⟶ 1 1 \longrightarrow\Gamma\cap N \longrightarrow\Gamma\longrightarrow\rho(\Gamma) \longrightarrow1 1 ⟶ Γ ∩ N ⟶ Γ ⟶ ρ ( Γ ) ⟶ 1 shows that Γ \Gamma Γ V i V_i V i B G BG B G cat A m ( G ) ≤ n \operatorname{cat}_{\mathrm{Am}}(G)\le n cat Am ( G ) ≤ n
Remark 5.7. The same argument applies more generally to a class of groups C \mathcal{C} C C \mathcal{C} C
We now apply Lemma 5.6 to graph products of pairs.
Proposition 5.8. Let L L L H i ⊴ G i H_i\trianglelefteq G_i H i ⊴ G i i i i Q i = G i / H i Q_i=G_i/H_i Q i = G i / H i
1 ⟶ ∏ i ∈ I H i ⟶ ∏ L ( G i , H i ) → ρ ∗ L Q i ⟶ 1. 1 \longrightarrow\prod_{i\in I} H_i \longrightarrow\prod_{L} (G_i,H_i) \xrightarrow{\rho} \mathop{*}_{L} Q_i \longrightarrow1. 1 ⟶ i ∈ I ∏ H i ⟶ L ∏ ( G i , H i ) ρ ∗ L Q i ⟶ 1. Proof. Recall that the graph product of pairs is obtained from the local groups
G J = ∏ j ∈ J G j × ∏ i ∉ J H i , J ∈ S ( L ) , G_J=\prod_{j\in J}G_j\times\prod_{i\notin J}H_i,\qquad J\in S(L), G J = j ∈ J ∏ G j × i ∈ / J ∏ H i , J ∈ S ( L ) , together with the natural inclusions associated with inclusions of simplices (see Definition 3.4). Since each H i H_i H i G i G_i G i N = ∏ i ∈ I H i N=\prod_{i\in I}H_i N = ∏ i ∈ I H i Π L 1 ( G i , H i ) \Pi_{L_1}(G_i,H_i) Π L 1 ( G i , H i ) G J / N ≅ ∏ j ∈ J Q j G_J/N\cong\prod_{j\in J}Q_j G J / N ≅ ∏ j ∈ J Q j Π L 1 ( G i , H i ) / N ≅ ∗ L 1 Q i \Pi_{L_1}(G_i,H_i)/N\cong\ast_{L_1}Q_i Π L 1 ( G i , H i ) / N ≅ ∗ L 1 Q i ∏ j ∈ J Q j \prod_{j\in J}Q_j ∏ j ∈ J Q j J ∈ S ( L ) J\in S(L) J ∈ S ( L ) L 1 L_1 L 1
Consequently, normal amenable pairs already give an improvement over the general estimates of the previous subsection.
Corollary 5.9. Under the hypotheses of Proposition 5.8 , assume in addition that every H i H_i H i
cat A m Π L 1 ( G i , H i ) ≤ cat A m ( ∗ L 1 Q i ) . \operatorname{cat}_{\mathrm{Am}}\Pi_{L_1}(G_i,H_i)\leq\operatorname{cat}_{\mathrm{Am}}\left(\ast_{L_1}Q_i\right). cat Am Π L 1 ( G i , H i ) ≤ cat Am ( ∗ L 1 Q i ) . Proof. The kernel ∏ i ∈ I H i \prod_{i\in I}H_i ∏ i ∈ I H i Proposition 5.8 is amenable, because the vertex set is finite and finite products of amenable groups are amenable. The result follows from Lemma 5.6 . □
In the split case the preceding inequality is an equality.
Lemma 5.10. Suppose that G i = H i ⋊ Q i G_i=H_i\rtimes Q_i G i = H i ⋊ Q i i i i ρ : Π L 1 ( G i , H i ) ⟶ ∗ L 1 Q i \rho:\Pi_{L_1}(G_i,H_i)\longrightarrow\ast_{L_1}Q_i ρ : Π L 1 ( G i , H i ) ⟶ ∗ L 1 Q i Proposition 5.8 admits a section. In particular, ∗ L 1 Q i \ast_{L_1}Q_i ∗ L 1 Q i Π L 1 ( G i , H i ) \Pi_{L_1}(G_i,H_i) Π L 1 ( G i , H i )
Proof. For every i i i s i : Q i ⟶ G i s_i:Q_i\longrightarrow G_i s i : Q i ⟶ G i { i , j } \{i,j\} { i , j } L 1 L_1 L 1 G i G_i G i G j G_j G j [ s i ( Q i ) , s j ( Q j ) ] = 1 [s_i(Q_i),s_j(Q_j)]=1 [ s i ( Q i ) , s j ( Q j )] = 1 s i s_i s i
s : ∗ L 1 Q i ⟶ Π L 1 ( G i , H i ) . s:\ast_{L_1}Q_i\longrightarrow\Pi_{L_1}(G_i,H_i). s : ∗ L 1 Q i ⟶ Π L 1 ( G i , H i ) . The composite ρ ∘ s \rho\circ s ρ ∘ s Q i Q_i Q i ρ ∘ s = id \rho\circ s=\operatorname{id} ρ ∘ s = id s s s
Theorem 5.11. Let L L L G i = H i ⋊ Q i G_i=H_i\rtimes Q_i G i = H i ⋊ Q i i i i H i H_i H i cat A m Π L 1 ( G i , H i ) = cat A m ( ∗ L 1 Q i ) \operatorname{cat}_{\mathrm{Am}}\Pi_{L_1}(G_i,H_i)=\operatorname{cat}_{\mathrm{Am}}(\ast_{L_1}Q_i) cat Am Π L 1 ( G i , H i ) = cat Am ( ∗ L 1 Q i )
Proof. By Proposition 5.8 , there is a short exact sequence
1 ⟶ ∏ i ∈ I H i ⟶ Π L 1 ( G i , H i ) ⟶ ∗ L 1 Q i ⟶ 1. 1\longrightarrow\prod_{i\in I}H_i\longrightarrow\Pi_{L_1}(G_i,H_i)\longrightarrow\ast_{L_1}Q_i\longrightarrow1. 1 ⟶ i ∈ I ∏ H i ⟶ Π L 1 ( G i , H i ) ⟶ ∗ L 1 Q i ⟶ 1. Its kernel is amenable, and hence Lemma 5.6 gives cat A m Π L 1 ( G i , H i ) ≤ cat A m ( ∗ L 1 Q i ) \operatorname{cat}_{\mathrm{Am}}\Pi_{L_1}(G_i,H_i)\leq\operatorname{cat}_{\mathrm{Am}}(\ast_{L_1}Q_i) cat Am Π L 1 ( G i , H i ) ≤ cat Am ( ∗ L 1 Q i ) Lemma 5.10 identifies ∗ L 1 Q i \ast_{L_1}Q_i ∗ L 1 Q i Π L 1 ( G i , H i ) \Pi_{L_1}(G_i,H_i) Π L 1 ( G i , H i ) cat A m ( ∗ L 1 Q i ) ≤ cat A m Π L 1 ( G i , H i ) \operatorname{cat}_{\mathrm{Am}}(\ast_{L_1}Q_i)\leq\operatorname{cat}_{\mathrm{Am}}\Pi_{L_1}(G_i,H_i) cat Am ( ∗ L 1 Q i ) ≤ cat Am Π L 1 ( G i , H i )
Theorem 5.11 reduces the amenable category of a large class of graph products of pairs to that of an ordinary graph product. Exact values for the latter were obtained by Li [19 Theorem 3.5] .
Theorem 5.12 (Li). Let Q Q Q F F F Q L Q_L Q L Q Q Q L L L
cat A m ( Q L ) = vcd ( W L ) , \operatorname{cat}_{\mathrm{Am}}(Q_L)=\operatorname{vcd}(W_L), cat Am ( Q L ) = vcd ( W L ) , where W L W_L W L L L L
Corollary 5.13. Let Q Q Q F F F i i i L L L G i = H i ⋊ Q G_i=H_i\rtimes Q G i = H i ⋊ Q H i H_i H i cat A m Π L 1 ( G i , H i ) = vcd ( W L ) \operatorname{cat}_{\mathrm{Am}}\Pi_{L_1}(G_i,H_i)=\operatorname{vcd}(W_L) cat Am Π L 1 ( G i , H i ) = vcd ( W L )
Proof. The quotient graph product is ∗ L 1 Q = Q L \ast_{L_1} Q = Q_L ∗ L 1 Q = Q L Theorem 5.11 gives
cat A m Π L 1 ( G i , H i ) = cat A m ( Q L ) . \operatorname{cat}_{\mathrm{Am}} \Pi_{L_1}(G_i,H_i)=\operatorname{cat}_{\mathrm{Am}}(Q_L). cat Am Π L 1 ( G i , H i ) = cat Am ( Q L ) . The conclusion follows from Theorem 5.12. □
Corollary 5.14. Let L L L G i = H i ⋊ Z G_i=H_i\rtimes\mathbb{Z} G i = H i ⋊ Z i i i H i H_i H i
cat A m Π L 1 ( G i , H i ) = vcd ( W L ) . \operatorname{cat}_{\mathrm{Am}}\Pi_{L_1}(G_i,H_i)=\operatorname{vcd}(W_L). cat Am Π L 1 ( G i , H i ) = vcd ( W L ) . Proof. This is Corollary 5.13 with Q = Z Q=\mathbb{Z} Q = Z A L = Z L A_L=\mathbb{Z}_L A L = Z L
The splitting hypothesis in Theorem 5.11 is used only to obtain the reverse inequality by embedding the quotient graph product into the graph product of pairs. Without a splitting, Corollary 5.9 still gives the upper bound, but equality is no longer automatic.
Question 5.15. Let
1 ⟶ N ⟶ G ⟶ Q ⟶ 1 1\longrightarrow N\longrightarrow G\longrightarrow Q\longrightarrow1 1 ⟶ N ⟶ G ⟶ Q ⟶ 1 be an extension with amenable kernel. Under what additional hypotheses does one have cat A m ( G ) = cat A m ( Q ) \operatorname{cat}_{\mathrm{Am}}(G)=\operatorname{cat}_{\mathrm{Am}}(Q) cat Am ( G ) = cat Am ( Q )
Join decompositions and product behaviour
Join decompositions play a fundamental role in the topology of polyhedral products. If a simplicial complex decomposes as a join, then the corresponding polyhedral product decomposes as a Cartesian product. In the setting of moment-angle manifolds, the relationship between join decompositions and product decompositions has been studied in particular by Amelotte and Briggs [1 ] . Here we record the corresponding behaviour for the graph products of pairs arising from polyhedral products and examine its interaction with the amenable-category computations of the previous subsection.
Let L = L ( 1 ) ∗ L ( 2 ) L=L^{(1)}*L^{(2)} L = L ( 1 ) ∗ L ( 2 ) I 1 I_1 I 1 I 2 I_2 I 2 L ( 1 ) L^{(1)} L ( 1 ) L ( 2 ) L^{(2)} L ( 2 ) I = I 1 ⊔ I 2 I=I_1\sqcup I_2 I = I 1 ⊔ I 2 r = 1 , 2 r=1,2 r = 1 , 2 ( A ( r ) , B ( r ) ) = { ( A i , B i ) } i ∈ I r (A^{(r)},B^{(r)})=\{(A_i,B_i)\}_{i\in I_r} ( A ( r ) , B ( r ) ) = {( A i , B i ) } i ∈ I r
Proposition 5.16. With the notation above, there is a natural homeomorphism
Z L ( A , B ) ≅ Z L ( 1 ) ( A ( 1 ) , B ( 1 ) ) × Z L ( 2 ) ( A ( 2 ) , B ( 2 ) ) . Z_L(A,B)\cong Z_{L^{(1)}}(A^{(1)},B^{(1)})\times Z_{L^{(2)}}(A^{(2)},B^{(2)}). Z L ( A , B ) ≅ Z L ( 1 ) ( A ( 1 ) , B ( 1 ) ) × Z L ( 2 ) ( A ( 2 ) , B ( 2 ) ) . Proof. Every simplex of the join L ( 1 ) ∗ L ( 2 ) L^{(1)}*L^{(2)} L ( 1 ) ∗ L ( 2 ) σ 1 ∪ σ 2 \sigma_1\cup\sigma_2 σ 1 ∪ σ 2 σ r ∈ L ( r ) \sigma_r\in L^{(r)} σ r ∈ L ( r )
∏ i ∈ σ 1 A i × ∏ i ∈ I 1 ∖ σ 1 B i × ∏ i ∈ σ 2 A i × ∏ i ∈ I 2 ∖ σ 2 B i . \prod_{i\in\sigma_1}A_i\times\prod_{i\in I_1\setminus\sigma_1}B_i\times\prod_{i\in\sigma_2}A_i\times\prod_{i\in I_2\setminus\sigma_2}B_i. i ∈ σ 1 ∏ A i × i ∈ I 1 ∖ σ 1 ∏ B i × i ∈ σ 2 ∏ A i × i ∈ I 2 ∖ σ 2 ∏ B i . Taking the union over all pairs ( σ 1 , σ 2 ) ∈ L ( 1 ) × L ( 2 ) (\sigma_1,\sigma_2)\in L^{(1)}\times L^{(2)} ( σ 1 , σ 2 ) ∈ L ( 1 ) × L ( 2 )
Z L ( 1 ) ∗ L ( 2 ) ( A , B ) = Z L ( 1 ) ( A ( 1 ) , B ( 1 ) ) × Z L ( 2 ) ( A ( 2 ) , B ( 2 ) ) . Z_{L^{(1)}*L^{(2)}}(A,B)=Z_{L^{(1)}}(A^{(1)},B^{(1)})\times Z_{L^{(2)}}(A^{(2)},B^{(2)}). Z L ( 1 ) ∗ L ( 2 ) ( A , B ) = Z L ( 1 ) ( A ( 1 ) , B ( 1 ) ) × Z L ( 2 ) ( A ( 2 ) , B ( 2 ) ) . The same decomposition is visible at the level of fundamental groups. In particular, in the transitive case it yields a direct product decomposition of graph products of pairs.
Proposition 5.17. Let L = L ( 1 ) ∗ L ( 2 ) L=L^{(1)}*L^{(2)} L = L ( 1 ) ∗ L ( 2 ) G = Π L 1 ( G i , H i ) G=\Pi_{L_1}(G_i,H_i) G = Π L 1 ( G i , H i ) r = 1 , 2 r=1,2 r = 1 , 2 G ( r ) = Π ( L ( r ) ) 1 ( G i , H i ) i ∈ I r G^{(r)}=\Pi_{(L^{(r)})_1}(G_i,H_i)_{i\in I_r} G ( r ) = Π ( L ( r ) ) 1 ( G i , H i ) i ∈ I r G ≅ G ( 1 ) × G ( 2 ) G\cong G^{(1)}\times G^{(2)} G ≅ G ( 1 ) × G ( 2 )
Proof. Since every vertex of L ( 1 ) L^{(1)} L ( 1 ) L ( 2 ) L^{(2)} L ( 2 ) I 1 I_1 I 1 I 2 I_2 I 2 Definition 3.4 ). Hence the canonical homomorphisms
G ( 1 ) ⟶ G , G ( 2 ) ⟶ G G^{(1)} \longrightarrow G,\qquad G^{(2)} \longrightarrow G G ( 1 ) ⟶ G , G ( 2 ) ⟶ G have commuting images and induce a homomorphism G ( 1 ) × G ( 2 ) ⟶ G G^{(1)} \times G^{(2)} \longrightarrow G G ( 1 ) × G ( 2 ) ⟶ G Π L 1 ( G i , H i ) \Pi_{L_1}(G_i,H_i) Π L 1 ( G i , H i ) I = I 1 ⊔ I 2 I=I_1\sqcup I_2 I = I 1 ⊔ I 2 G ⟶ G ( 1 ) × G ( 2 ) G \longrightarrow G^{(1)} \times G^{(2)} G ⟶ G ( 1 ) × G ( 2 )
Remark 5.18. Thus a join decomposition is detected simultaneously at three levels: L = L ( 1 ) ∗ L ( 2 ) L=L^{(1)}*L^{(2)} L = L ( 1 ) ∗ L ( 2 ) Z L ( A , B ) ≅ Z L ( 1 ) ( A ( 1 ) , B ( 1 ) ) × Z L ( 2 ) ( A ( 2 ) , B ( 2 ) ) Z_L(A,B)\cong Z_{L^{(1)}}(A^{(1)},B^{(1)})\times Z_{L^{(2)}}(A^{(2)},B^{(2)}) Z L ( A , B ) ≅ Z L ( 1 ) ( A ( 1 ) , B ( 1 ) ) × Z L ( 2 ) ( A ( 2 ) , B ( 2 ) ) Π L 1 ( G i , H i ) ≅ G ( 1 ) × G ( 2 ) \Pi_{L_1}(G_i,H_i)\cong G^{(1)}\times G^{(2)} Π L 1 ( G i , H i ) ≅ G ( 1 ) × G ( 2 )
We next combine Proposition 5.17 with the exact computation from Theorem 5.11 . The resulting behaviour is more subtle than ordinary additivity.
Corollary 5.19. Suppose that L = L ( 1 ) ∗ L ( 2 ) L=L^{(1)}*L^{(2)} L = L ( 1 ) ∗ L ( 2 ) G i = H i ⋊ Q i G_i=H_i\rtimes Q_i G i = H i ⋊ Q i i i i H i H_i H i
cat A m Π L 1 ( G i , H i ) = cat A m ( ∗ L 1 Q i ) . \operatorname{cat}_{\mathrm{Am}}\Pi_{L_1}(G_i,H_i)=\operatorname{cat}_{\mathrm{Am}}\left(\underset{L_1}{*} Q_i\right). cat Am Π L 1 ( G i , H i ) = cat Am ( L 1 ∗ Q i ) . Moreover,
∗ L 1 Q i ≅ ( ∗ ( L ( 1 ) ) 1 Q i ) × ( ∗ ( L ( 2 ) ) 1 Q i ) . \underset{L_1}{*}Q_i\cong\left(\underset{(L^{(1)})_1}{*}Q_i\right)\times\left(\underset{(L^{(2)})_1}{*}Q_i\right). L 1 ∗ Q i ≅ ( ( L ( 1 ) ) 1 ∗ Q i ) × ( ( L ( 2 ) ) 1 ∗ Q i ) . Proof. The first assertion is Theorem 5.11 . For the second, every vertex of L ( 1 ) L^{(1)} L ( 1 ) L ( 2 ) L^{(2)} L ( 2 )
When all quotient groups coincide with a group to which Li’s theorem applies, the preceding statement gives an exact combinatorial description.
Corollary 5.20. Let Q Q Q F F F G i = H i ⋊ Q G_i=H_i\rtimes Q G i = H i ⋊ Q i i i H i H_i H i L = L ( 1 ) ∗ L ( 2 ) L=L^{(1)}*L^{(2)} L = L ( 1 ) ∗ L ( 2 ) cat A m Π L 1 ( G i , H i ) = vcd ( W L ) \operatorname{cat}_{\mathrm{Am}}\Pi_{L_1}(G_i,H_i)=\operatorname{vcd}(W_L) cat Am Π L 1 ( G i , H i ) = vcd ( W L ) W L ≅ W L ( 1 ) × W L ( 2 ) W_L\cong W_{L^{(1)}}\times W_{L^{(2)}} W L ≅ W L ( 1 ) × W L ( 2 ) cat A m Π L 1 ( G i , H i ) = vcd ( W L ( 1 ) × W L ( 2 ) ) \operatorname{cat}_{\mathrm{Am}}\Pi_{L_1}(G_i,H_i)=\operatorname{vcd}\left(W_{L^{(1)}}\times W_{L^{(2)}}\right) cat Am Π L 1 ( G i , H i ) = vcd ( W L ( 1 ) × W L ( 2 ) )
Proof. By Corollary 5.13,
cat A m Π L 1 ( G i , H i ) = vcd ( W L ) . \operatorname{cat}_{\mathrm{Am}}\Pi_{L_1}(G_i,H_i)=\operatorname{vcd}(W_L). cat Am Π L 1 ( G i , H i ) = vcd ( W L ) . The defining graph of the right-angled Coxeter group associated with a join is itself a join, and therefore W L ( 1 ) ∗ L ( 2 ) ≅ W L ( 1 ) × W L ( 2 ) W_{L^{(1)}*L^{(2)}}\cong W_{L^{(1)}}\times W_{L^{(2)}} W L ( 1 ) ∗ L ( 2 ) ≅ W L ( 1 ) × W L ( 2 )
Remark 5.21. It is important that the corollary 5.20 is formulated in terms of vcd ( W L ( 1 ) × W L ( 2 ) ) \operatorname{vcd}\left(W_{L^{(1)}}\times W_{L^{(2)}}\right) vcd ( W L ( 1 ) × W L ( 2 ) ) [19 ] exhibits right-angled Artin groups for which
cat A m ( A L ( 1 ) × A L ( 2 ) ) < cat A m ( A L ( 1 ) ) + cat A m ( A L ( 2 ) ) . \operatorname{cat}_{\mathrm{Am}}(A_{L^{(1)}}\times A_{L^{(2)}})<\operatorname{cat}_{\mathrm{Am}}(A_{L^{(1)}})+\operatorname{cat}_{\mathrm{Am}}(A_{L^{(2)}}). cat Am ( A L ( 1 ) × A L ( 2 ) ) < cat Am ( A L ( 1 ) ) + cat Am ( A L ( 2 ) ) . Thus join decompositions of the defining simplicial complex give canonical product decompositions of both the polyhedral product and its graph-product-of-pairs fundamental group, while the numerical behaviour of amenable category under these products can be genuinely non-additive.
The preceding discussion mirrors a familiar phenomenon in toric topology: combinatorial join decompositions of the defining simplicial complex are reflected geometrically by Cartesian product decompositions of the associated polyhedral product. The results above show that, for the aspherical polyhedral products considered here, the same decomposition is also visible at the level of the relative fundamental group. In contrast with the geometric product decomposition, however, amenable category need not decompose additively. This provides a group-theoretic complement to the join–product viewpoint appearing in the study of product decompositions of moment-angle manifolds.
Simplicial volume and manifold applications
We now turn the amenable-category estimates obtained above into vanishing results for simplicial volume. An important point is that, at this stage, the manifold itself need not be aspherical: it is enough to control the amenable category of its fundamental group.
Recall that if M M M m m m simplicial volume ∥ M ∥ \lVert M\rVert ∥ M ∥ ℓ 1 \ell^1 ℓ 1 [ M ] ∈ H m ( M ; R ) [M] \in H_m(M;\mathbb{R}) [ M ] ∈ H m ( M ; R )
Theorem 5.22 (Gromov [15 ] ). Let M M M cat A m ( M ) < dim M \operatorname{cat}_{\mathrm{Am}}(M) < \dim M cat Am ( M ) < dim M ∥ M ∥ = 0 \lVert M\rVert= 0 ∥ M ∥ = 0
Lemma 5.23. Let X X X C W CW C W cat A m ( X ) ≤ cat A m ( π 1 ( X ) ) \operatorname{cat}_{\mathrm{Am}}(X) \le\operatorname{cat}_{\mathrm{Am}}(\pi_1(X)) cat Am ( X ) ≤ cat Am ( π 1 ( X ))
Proof. Let c X : X ⟶ B π 1 ( X ) c_X:X \longrightarrow B\pi_1(X) c X : X ⟶ B π 1 ( X ) B π 1 ( X ) = U 0 ∪ ⋯ ∪ U n B\pi_1(X)=U_0\cup\cdots\cup U_n B π 1 ( X ) = U 0 ∪ ⋯ ∪ U n n = cat A m ( π 1 ( X ) ) n=\operatorname{cat}_{\mathrm{Am}}(\pi_1(X)) n = cat Am ( π 1 ( X )) V i = c X − 1 ( U i ) V_i=c_X^{-1}(U_i) V i = c X − 1 ( U i ) V i V_i V i X X X
For every path component V V V V i V_i V i π 1 ( V ) ⟶ π 1 ( X ) \pi_1(V)\longrightarrow\pi_1(X) π 1 ( V ) ⟶ π 1 ( X ) U i U_i U i π 1 ( X ) \pi_1(X) π 1 ( X ) V i V_i V i X X X i i i cat A m ( X ) ≤ n \operatorname{cat}_{\mathrm{Am}}(X)\le n cat Am ( X ) ≤ n
Proposition 5.24. Let M M M m m m cat A m ( π 1 ( M ) ) < m \operatorname{cat}_{\mathrm{Am}}(\pi_1(M))<m cat Am ( π 1 ( M )) < m ∥ M ∥ = 0 \lVert M\rVert=0 ∥ M ∥ = 0
Proof. By Lemma 5.23 ,
cat A m ( M ) ≤ cat A m ( π 1 ( M ) ) < m . \operatorname{cat}_{\mathrm{Am}}(M)\le\operatorname{cat}_{\mathrm{Am}}(\pi_1(M))<m. cat Am ( M ) ≤ cat Am ( π 1 ( M )) < m . The conclusion follows from Theorem 5.22 . □
Theorem 5.25. Let M M M m m m π 1 ( M ) ≅ Π L ( G i , H i ) \pi_1(M)\cong\Pi_L(G_i,H_i) π 1 ( M ) ≅ Π L ( G i , H i ) L L L J ∈ S ( L ) J\in S(L) J ∈ S ( L ) G J = ∏ j ∈ J G j × ∏ i ∉ J H i G_J=\prod_{j\in J}G_j\times\prod_{i\notin J}H_i G J = ∏ j ∈ J G j × ∏ i ∈ / J H i
max { cat A m ( ∏ i ∈ I H i ) , max ∅ ≠ J ∈ S ( L ) ( gd ( G J ) + ∣ J ∣ ) } < m , \max\left\{\operatorname{cat}_{\mathrm{Am}}\left(\prod_{i\in I}H_i\right),\ \max_{\varnothing\ne J\in S(L)}\left(\operatorname{gd}(G_J)+\lvert J\rvert\right)\right\}<m, max { cat Am ( i ∈ I ∏ H i ) , ∅ = J ∈ S ( L ) max ( gd ( G J ) + ∣ J ∣ ) } < m , then ∥ M ∥ = 0 \lVert M\rVert=0 ∥ M ∥ = 0
Proof. Corollary 5.2 gives
cat A m ( π 1 ( M ) ) ≤ max { cat A m ( ∏ i ∈ I H i ) , max ∅ ≠ J ∈ S ( L ) ( gd ( G J ) + ∣ J ∣ ) } . \operatorname{cat}_{\mathrm{Am}}(\pi_1(M))\le\max\left\{\operatorname{cat}_{\mathrm{Am}}\left(\prod_{i\in I}H_i\right),\ \max_{\varnothing\ne J\in S(L)}\left(\operatorname{gd}(G_J)+\lvert J\rvert\right)\right\}. cat Am ( π 1 ( M )) ≤ max { cat Am ( i ∈ I ∏ H i ) , ∅ = J ∈ S ( L ) max ( gd ( G J ) + ∣ J ∣ ) } . The assumed strict inequality and Proposition 5.24 give the claim. □
Corollary 5.26. Let M M M m m m π 1 ( M ) ≅ Π L ( G i , H i ) \pi_1(M)\cong\Pi_L(G_i,H_i) π 1 ( M ) ≅ Π L ( G i , H i ) L L L G i G_i G i dim L + 1 < m \dim L+1<m dim L + 1 < m ∥ M ∥ = 0 \lVert M\rVert=0 ∥ M ∥ = 0
Proof. By Corollary 5.4 , cat A m ( π 1 ( M ) ) ≤ dim L + 1 < m \operatorname{cat}_{\mathrm{Am}}(\pi_1(M))\le\dim L+1<m cat Am ( π 1 ( M )) ≤ dim L + 1 < m Proposition 5.24 . □
The exact computations of Subsection 5.2 yield a sharper criterion for manifolds whose fundamental groups are split graph products of pairs.
Theorem 5.27. Let M M M m m m π 1 ( M ) ≅ Π L 1 ( H i ⋊ Q i , H i ) \pi_1(M) \cong\mathop{\Pi}_{L_1}(H_i \rtimes Q_i,H_i) π 1 ( M ) ≅ Π L 1 ( H i ⋊ Q i , H i ) H i H_i H i cat A m ( ∗ L 1 Q i ) < m \operatorname{cat}_{\mathrm{Am}}(*_{L_1} Q_i)<m cat Am ( ∗ L 1 Q i ) < m ∥ M ∥ = 0 \lVert M\rVert=0 ∥ M ∥ = 0
Proof. By Theorem 5.11 ,
cat A m ( π 1 ( M ) ) = cat A m ( ∗ L 1 Q i ) . \operatorname{cat}_{\mathrm{Am}}(\pi_1(M))=\operatorname{cat}_{\mathrm{Am}}\left(*_{L_1}Q_i\right). cat Am ( π 1 ( M )) = cat Am ( ∗ L 1 Q i ) . The result follows from Proposition 5.24 .
Corollary 5.28. Let Q Q Q F F F M M M m m m π 1 ( M ) ≅ Π L 1 ( H i ⋊ Q , H i ) \pi_1(M)\cong\mathop{\Pi}_{L_1}(H_i\rtimes Q,H_i) π 1 ( M ) ≅ Π L 1 ( H i ⋊ Q , H i ) H i H_i H i vcd ( W L ) < m \operatorname{vcd}(W_L)<m vcd ( W L ) < m ∥ M ∥ = 0 \lVert M\rVert=0 ∥ M ∥ = 0
Proof. By Corollary 5.13, cat A m ( π 1 ( M ) ) = vcd ( W L ) \operatorname{cat}_{\mathrm{Am}}(\pi_1(M))=\operatorname{vcd}(W_L) cat Am ( π 1 ( M )) = vcd ( W L )
cat A m ( π 1 ( M ) ) < m , \operatorname{cat}_{\mathrm{Am}}(\pi_1(M))<m, cat Am ( π 1 ( M )) < m , and Proposition 5.24 applies.
Corollary 5.29. Let M M M m m m π 1 ( M ) ≅ Π L 1 ( H i ⋊ Z , H i ) \pi_1(M)\cong\mathop{\Pi}_{L_1}(H_i\rtimes\mathbb{Z},H_i) π 1 ( M ) ≅ Π L 1 ( H i ⋊ Z , H i ) H i H_i H i vcd ( W L ) < m \operatorname{vcd}(W_L)<m vcd ( W L ) < m ∥ M ∥ = 0 \lVert M\rVert=0 ∥ M ∥ = 0
Remark 5.30. The group-theoretic vanishing criteria above should be distinguished from the classical vanishing results coming from continuous group actions. For example, whenever a complex moment-angle complex Z L = Z L ( D 2 , S 1 ) Z_L=Z_L(D^2,S^1) Z L = Z L ( D 2 , S 1 )
The real moment-angle case is different. A real moment-angle manifold Z L R = Z L ( D 1 , S 0 ) Z_L^{\mathbb{R}}=Z_L(D^1,S^0) Z L R = Z L ( D 1 , S 0 ) ( Z / 2 ) m (\mathbb{Z}/2)^m ( Z /2 ) m L L L dim Z L R = dim L + 1 \dim Z_L^{\mathbb{R}}=\dim L+1 dim Z L R = dim L + 1 Corollary 5.26 does not yield vanishing, since its hypothesis requires the strict inequality dim L + 1 < dim M \dim L+1<\dim M dim L + 1 < dim M
Remark 5.31. A natural testing ground for the preceding criteria is provided by manifolds obtained from Davis’ reflection-group construction. Their simplicial volume is not uniformly zero, and hence no blanket vanishing statement can hold for this class. It would therefore be interesting to determine whether the amenable-category estimates above single out subclasses of Davis-type manifolds for which the simplicial volume vanishes.
Polyhedral products of classifying spaces
The preceding results become particularly concrete for polyhedral products built from classifying spaces of pairs of groups. Let G i G_i G i H i ≤ G i H_i\leq G_i H i ≤ G i i ∈ I i\in I i ∈ I B H i BH_i B H i B G i BG_i B G i H i ↪ G i H_i\hookrightarrow G_i H i ↪ G i B H i ↪ B G i BH_i\hookrightarrow BG_i B H i ↪ B G i B H i → B G i BH_i\to BG_i B H i → B G i Z L ( B G , B H ) = Z L ( ( B G i , B H i ) i ∈ I ) Z_L(BG,BH)=Z_L((BG_i,BH_i)_{i\in I}) Z L ( B G , B H ) = Z L (( B G i , B H i ) i ∈ I )
The advantage of this class is that both the fundamental-group data and the stabilizers of the associated building admit purely group-theoretic descriptions.
The pairs ( B G i , B H i ) (BG_i,BH_i) ( B G i , B H i )
Let p i : E G i ⟶ B G i p_i:EG_i\longrightarrow BG_i p i : E G i ⟶ B G i p i − 1 ( B H i ) p_i^{-1}(BH_i) p i − 1 ( B H i ) G i G_i G i H i H_i H i π 0 ( p i − 1 ( B H i ) ) ≅ G i / H i \pi_0(p_i^{-1}(BH_i))\cong G_i/H_i π 0 ( p i − 1 ( B H i )) ≅ G i / H i G i G_i G i ( B G i , B H i ) (BG_i,BH_i) ( B G i , B H i ) ( G i , G i / H i ) (G_i,G_i/H_i) ( G i , G i / H i )
Proposition 6.1. Let L L L I I I H i ≤ G i H_i\leq G_i H i ≤ G i i ∈ I i\in I i ∈ I
(1) the fundamental-group data associated with ( B G i , B H i ) (BG_i,BH_i) ( B G i , B H i ) E i ≅ G i / H i E_i\cong G_i/H_i E i ≅ G i / H i
(2) there is a natural isomorphism π 1 Z L ( B G , B H ) ≅ Π L 1 ( G i , H i ) \pi_1Z_L(BG,BH)\cong\Pi_{L_1}(G_i,H_i) π 1 Z L ( B G , B H ) ≅ Π L 1 ( G i , H i )
(3) if H i < G i H_i<G_i H i < G i i ∈ I i\in I i ∈ I Z L ( B G , B H ) Z_L(BG,BH) Z L ( B G , B H ) L L L
Proof. The universal cover of B G i BG_i B G i E G i EG_i E G i B H i ⟶ B G i BH_i\longrightarrow BG_i B H i ⟶ B G i H i ↪ G i H_i\hookrightarrow G_i H i ↪ G i p i − 1 ( B H i ) p_i^{-1}(BH_i) p i − 1 ( B H i ) B H i BH_i B H i
The deck-transformation action of G i G_i G i H i H_i H i E i = π 0 ( p i − 1 ( B H i ) ) ≅ G i / H i E_i=\pi_0(p_i^{-1}(BH_i))\cong G_i/H_i E i = π 0 ( p i − 1 ( B H i )) ≅ G i / H i
By Theorem 3.7 , π 1 Z L ( B G , B H ) ≅ Z L 1 ( G , E ) \pi_1Z_L(BG,BH)\cong Z_{L_1}(G,E) π 1 Z L ( B G , B H ) ≅ Z L 1 ( G , E ) G i ↷ G i / H i G_i\curvearrowright G_i/H_i G i ↷ G i / H i Z L 1 ( G , E ) ≅ Π L 1 ( G i , H i ) Z_{L_1}(G,E)\cong\Pi_{L_1}(G_i,H_i) Z L 1 ( G , E ) ≅ Π L 1 ( G i , H i )
Finally, suppose that every H i H_i H i G i G_i G i ∣ G i / H i ∣ > 1 |G_i/H_i|>1 ∣ G i / H i ∣ > 1 i i i B G i BG_i B G i B H i BH_i B H i B H i ⟶ B G i BH_i\longrightarrow BG_i B H i ⟶ B G i π 1 \pi_1 π 1 ( B G i , B H i ) (BG_i,BH_i) ( B G i , B H i ) L L L
Remark 6.2. The properness assumption in Proposition 6.1 (3) is used only to guarantee that ∣ G i / H i ∣ > 1 |G_i/H_i|>1 ∣ G i / H i ∣ > 1 i i i H i = G i H_i=G_i H i = G i G i G_i G i I ′ ′ = { i ∈ I ∣ H i ≠ G i } I^{\prime\prime}=\{i\in I\mid H_i\ne G_i\} I ′′ = { i ∈ I ∣ H i = G i } L ′ ′ = L I ′ ′ L^{\prime\prime}=L_{I^{\prime\prime}} L ′′ = L I ′′ L ′ ′ L^{\prime\prime} L ′′
When L L L H i H_i H i Π L 1 ( G i , H i ) \Pi_{L_1}(G_i,H_i) Π L 1 ( G i , H i ) J ∈ S ( L ) J\in S(L) J ∈ S ( L ) G J = ∏ j ∈ J G j × ∏ i ∈ I ∖ J H i G_J=\prod_{j\in J}G_j\times\prod_{i\in I\setminus J}H_i G J = ∏ j ∈ J G j × ∏ i ∈ I ∖ J H i Proposition 3.17 , the stabilizer of a cube of type J J J G J G_J G J G ∅ = ∏ i ∈ I H i G_{\varnothing}=\prod_{i\in I}H_i G ∅ = ∏ i ∈ I H i
We therefore obtain directly from Theorem 4.9 the following form of the main estimate.
Corollary 6.3. Let L L L H i < G i H_i<G_i H i < G i i ∈ I i\in I i ∈ I Z = Z L ( B G , B H ) Z=Z_L(BG,BH) Z = Z L ( B G , B H ) F \mathcal{F} F G = π 1 ( Z ) G=\pi_1(Z) G = π 1 ( Z )
cat F ( G ) ≤ max { cat F ∣ G ∅ ( G ∅ ) , max ∅ ≠ J ∈ S ( L ) ( gd ( G J ) + ∣ J ∣ ) } . \operatorname{cat}_{\mathcal{F}}(G)\leq\max\left\{\operatorname{cat}_{\mathcal{F}|G_{\varnothing}}(G_{\varnothing}),\ \max_{\varnothing\ne J\in S(L)}\left(\operatorname{gd}(G_J)+|J|\right)\right\}. cat F ( G ) ≤ max { cat F ∣ G ∅ ( G ∅ ) , ∅ = J ∈ S ( L ) max ( gd ( G J ) + ∣ J ∣ ) } . where G ∅ = ∏ i ∈ I H i G_{\varnothing}=\prod_{i\in I}H_i G ∅ = ∏ i ∈ I H i G J = ∏ j ∈ J G j × ∏ i ∉ J H i G_J=\prod_{j\in J}G_j\times\prod_{i\notin J}H_i G J = ∏ j ∈ J G j × ∏ i ∈ / J H i
Proof. By Proposition 6.1 , G ≅ Π L 1 ( G i , H i ) G\cong\Pi_{L_1}(G_i,H_i) G ≅ Π L 1 ( G i , H i )
Example 6.4. If H i = { 1 } H_i=\{1\} H i = { 1 } i i i B H i = ∗ BH_i=* B H i = ∗ Z L ( B G , ∗ ) Z_L(BG,*) Z L ( B G , ∗ ) L L L π 1 Z L ( B G , ∗ ) ≅ ∗ L 1 G i \pi_1Z_L(BG,*)\cong*_ {L_1}G_i π 1 Z L ( B G , ∗ ) ≅ ∗ L 1 G i G J = ∏ j ∈ J G j G_J=\prod_{j\in J}G_j G J = ∏ j ∈ J G j
Example 6.5. Taking G i = Z G_i=\mathbb{Z} G i = Z H i = { 1 } H_i=\{1\} H i = { 1 } Z L ( S 1 , ∗ ) ≃ B A L 1 Z_L(S^1,*)\simeq BA_{L_1} Z L ( S 1 , ∗ ) ≃ B A L 1 L L L A L 1 A_{L_1} A L 1
Example 6.6 (Real Davis–Januszkiewicz space). Taking G i = Z / 2 G_i = \mathbb{Z}/2 G i = Z /2 H i = { 1 } H_i = \{1\} H i = { 1 } i ∈ I i \in I i ∈ I Z L ( R P ∞ , ∗ ) Z_L(\mathbb{R}P^\infty,*) Z L ( R P ∞ , ∗ ) L L L Proposition 6.1 implies that this space is aspherical, and
π 1 Z L ( R P ∞ , ∗ ) ≅ L 1 ∗ Z / 2 = W L 1 . \pi_1 Z_L(\mathbb{R}P^\infty,*) \cong{}^{*}_{L_1}\mathbb{Z}/2 = W_{L_1}. π 1 Z L ( R P ∞ , ∗ ) ≅ L 1 ∗ Z /2 = W L 1 . where W L 1 W_{L_1} W L 1 L L L Z L ( R P ∞ , ∗ ) ≃ B W L 1 Z_L(\mathbb{R}P^\infty,*) \simeq BW_{L_1} Z L ( R P ∞ , ∗ ) ≃ B W L 1 L L L
Farrell–Jones inheritance and rigidity consequences
The preceding sections describe the fundamental groups of polyhedral products in terms of relative graph products and graph products of pairs, and provide contractible right-angled buildings on which these groups act. We now investigate the consequences of this structure for the Farrell–Jones conjecture and for topological rigidity.
The relevant framework is the Farrell–Jones conjecture in algebraic K K K L L L
Our purpose in this section is threefold. First, we recall the Farrell–Jones theorem of Gandini–Rüping for ordinary graph products [GR13]. Second, we develop a pair-deletion argument for graph products of pairs, reducing the global Farrell–Jones problem to a local combination problem associated with the inclusions H i ≤ G i H_i \leq G_i H i ≤ G i
Manifold polyhedral products
We shall call a polyhedral product M = Z L ( A , B ) M = Z_L(A,B) M = Z L ( A , B ) manifold polyhedral product if it is a topological manifold. In this section we are primarily interested in the case in which M M M
If M = Z L ( A , B ) M = Z_L(A,B) M = Z L ( A , B ) M ≃ B π 1 ( M ) M \simeq B\pi_1(M) M ≃ B π 1 ( M ) M M M Theorem 3.7 , this group has the description
π 1 ( M ) ≅ Z L 1 ( G , E ) , \pi_1(M) \cong Z_{L_1}(G,E), π 1 ( M ) ≅ Z L 1 ( G , E ) , where
G i = π 1 ( A ( i ) ) , E i = π 0 ( B ( i ) ) . G_i = \pi_1(A(i)), \qquad E_i = \pi_0(B(i)). G i = π 1 ( A ( i )) , E i = π 0 ( B ( i )) . Thus the rigidity problem for an aspherical manifold polyhedral product naturally becomes a rigidity problem for a relative graph product.
Remark 7.1. If M M M G = π 1 ( M ) G = \pi_1(M) G = π 1 ( M ) M M M cd ( G ) = dim M \operatorname{cd}(G) = \dim M cd ( G ) = dim M
The building description developed in Subsection 3.3 gives an additional geometric structure. Under the flag hypotheses considered above, G ↷ B L G \curvearrowright B_L G ↷ B L
Remark 7.2. The action G ↷ B L G \curvearrowright B_L G ↷ B L J J J
G J = ∏ j ∈ J G j × ∏ i ∉ J H i , G_J = \prod_{j \in J} G_j \times\prod_{i \notin J} H_i, G J = j ∈ J ∏ G j × i ∈ / J ∏ H i , which can be infinite. Hence the fact that B L B_L B L G G G
This distinction becomes essential when applying the Farrell–Jones conjecture. A basic example is provided by real moment-angle manifolds. If L L L Z L R = Z L ( D 1 , S 0 ) Z_L^{\mathbb{R}}=Z_L(D^1,S^0) Z L R = Z L ( D 1 , S 0 ) W L W_L W L W L W_L W L [13 ] .
Farrell–Jones
We briefly recall the form of the Farrell–Jones conjecture needed below. Let V C y c \mathcal{VCyc} V C y c G G G
H n G ( E V C y c G ; K A ) ⟶ K n ( A ⋊ G ) H_n^G(E_{\mathcal{VCyc}}G;\mathbf{K}_{\mathcal{A}})\longrightarrow K_n(\mathcal{A}\rtimes G) H n G ( E V C y c G ; K A ) ⟶ K n ( A ⋊ G ) and
H n G ( E V C y c G ; L A ⟨ − ∞ ⟩ ) ⟶ L n ⟨ − ∞ ⟩ ( A ⋊ G ) H_n^G(E_{\mathcal{VCyc}}G;\mathbf{L}^{\langle-\infty\rangle}_{\mathcal{A}})\longrightarrow L_n^{\langle-\infty\rangle}(\mathcal{A}\rtimes G) H n G ( E V C y c G ; L A ⟨ − ∞ ⟩ ) ⟶ L n ⟨ − ∞ ⟩ ( A ⋊ G ) are isomorphisms, for suitable additive categories A \mathcal{A} A G G G K K K L L L
The relationship between graph products and assembly maps in algebraic K K K L L L K K K L L L K K K
This local-to-global structure is closely related to the building description used in the present paper: for a simplex J ∈ S ( L ) J\in S(L) J ∈ S ( L ) G J = ∏ j ∈ J G j G_J=\prod_{j\in J}G_j G J = ∏ j ∈ J G j K K K L L L
We shall use the version with coefficients in additive categories and finite wreath products. Following Gandini–Rüping [GR13], let C F J \mathrm{CFJ} CFJ K K K L L L
The class C F J \mathrm{CFJ} CFJ C A T ( 0 ) \mathrm{CAT}(0) CAT ( 0 ) f : G ⟶ Q f:G\longrightarrow Q f : G ⟶ Q Q ∈ C F J Q\in\mathrm{CFJ} Q ∈ CFJ ker ( f ) ∈ C F J \ker(f)\in\mathrm{CFJ} ker ( f ) ∈ CFJ f − 1 ( C ) ∈ C F J f^{-1}(C)\in\mathrm{CFJ} f − 1 ( C ) ∈ CFJ C ≤ Q C\leq Q C ≤ Q G ∈ C F J G\in\mathrm{CFJ} G ∈ CFJ
The main result of Gandini–Rüping relevant to the present paper is the following.
Theorem 7.3 (Gandini–Rüping [GR13, Theorem 1]). Let I I I ( G i ) i ∈ I (G_i)_{i\in I} ( G i ) i ∈ I ∗ I G i *_{I}G_i ∗ I G i
Theorem 7.3 gives an immediate Farrell–Jones result for those aspherical polyhedral products whose fundamental groups are ordinary graph products. We specialize this theorem to our setting.
Proposition 7.4. Let Z = Z L ( A , B ) Z=Z_L(A,B) Z = Z L ( A , B ) G i ↷ E i G_i\curvearrowright E_i G i ↷ E i π 1 ( Z ) ≅ ∗ L 1 G i \pi_1(Z)\cong*_{L_1}G_i π 1 ( Z ) ≅ ∗ L 1 G i G i G_i G i π 1 ( Z ) \pi_1(Z) π 1 ( Z )
Proof. By Proposition 3.15, the simple transitivity hypothesis identifies the relative graph product with the ordinary graph product π 1 ( Z ) ≅ ∗ L 1 G i \pi_1(Z)\cong*_{L_1}G_i π 1 ( Z ) ≅ ∗ L 1 G i □ \square □
For the classifying-space case, this gives:
Corollary 7.5. Let L L L ( G i ) i ∈ I (G_i)_{i\in I} ( G i ) i ∈ I C F J \mathcal{C}_{\mathrm{FJ}} C FJ π 1 Z L ( B G , ∗ ) ≅ ∗ L G i \pi_1 Z_L(BG,\ast)\cong\ast_L G_i π 1 Z L ( B G , ∗ ) ≅ ∗ L G i C F J \mathcal{C}_{\mathrm{FJ}} C FJ
The situation is subtler for graph products of pairs. If E i ≅ G i / H i E_i\cong G_i/H_i E i ≅ G i / H i π 1 ( Z ) ≅ Π L ( G i , H i ) \pi_1(Z)\cong\Pi_L(G_i,H_i) π 1 ( Z ) ≅ Π L ( G i , H i ) Theorem 7.3 does not apply directly.
Pair deletion
Let Γ \Gamma Γ P Γ = Π Γ ( G i , H i ) P_\Gamma=\Pi_\Gamma(G_i,H_i) P Γ = Π Γ ( G i , H i ) v ∈ V ( Γ ) v\in V(\Gamma) v ∈ V ( Γ )
Γ ′ = Γ − { v } , P ′ = Π Γ ′ ( G i , H i ) . \Gamma'=\Gamma-\{v\},\qquad P'=\Pi_{\Gamma'}(G_i,H_i). Γ ′ = Γ − { v } , P ′ = Π Γ ′ ( G i , H i ) . Recall that lk Γ ( v ) = { i ≠ v : { i , v } ∈ E ( Γ ) } \operatorname{lk}_{\Gamma}(v)=\{i\ne v:\{i,v\}\in E(\Gamma)\} lk Γ ( v ) = { i = v : { i , v } ∈ E ( Γ )} st Γ ( v ) = { v } ∪ lk Γ ( v ) \operatorname{st}_{\Gamma}(v)=\{v\}\cup\operatorname{lk}_{\Gamma}(v) st Γ ( v ) = { v } ∪ lk Γ ( v )
Definition 7.6. The relative link subgroup of v v v
R v = ⟨ G i ( i ∈ lk Γ ( v ) ) , H i ( i ∉ st Γ ( v ) ) ⟩ ≤ P ′ . R_v=\langle G_i\ (i\in\operatorname{lk}_{\Gamma}(v)),\ H_i\ (i\notin\operatorname{st}_{\Gamma}(v))\rangle\le P'. R v = ⟨ G i ( i ∈ lk Γ ( v )) , H i ( i ∈ / st Γ ( v ))⟩ ≤ P ′ . When all H i H_i H i R v R_v R v v v v
Proposition 7.7 (Pair-deletion decomposition). There is a natural isomorphism
P Γ ≅ ( H v × P ′ ) ∗ H v × R v ( G v × R v ) . P_\Gamma\cong(H_v\times P')\ast_{H_v\times R_v}(G_v\times R_v). P Γ ≅ ( H v × P ′ ) ∗ H v × R v ( G v × R v ) . Proof. Recall that the graph product of pairs is the colimit of the system of local groups G J = ∏ j ∈ J G j × ∏ i ∉ J H i G_J=\prod_{j\in J}G_j\times\prod_{i\notin J}H_i G J = ∏ j ∈ J G j × ∏ i ∈ / J H i J J J Γ \Gamma Γ Definition 3.4 ). Every simplex containing v v v { v } ∪ σ \{v\}\cup\sigma { v } ∪ σ σ ⊆ lk Γ ( v ) \sigma\subseteq\operatorname{lk}_{\Gamma}(v) σ ⊆ lk Γ ( v ) v v v G v G_v G v R v R_v R v
By construction, G v G_v G v R v R_v R v H v H_v H v P ′ P' P ′
A = H v × P ′ , B = G v × R v A=H_v\times P',\qquad B=G_v\times R_v A = H v × P ′ , B = G v × R v have common subgroup E = H v × R v E=H_v\times R_v E = H v × R v P Γ P_\Gamma P Γ A A A B B B E E E
P Γ ≅ A ∗ E B . P_\Gamma\cong A\ast_E B. P Γ ≅ A ∗ E B . The preceding decomposition defines a canonical epimorphism ρ v : P Γ ⟶ P ′ \rho_v:P_\Gamma\longrightarrow P' ρ v : P Γ ⟶ P ′
ρ v ( h , p ) = p ( h , p ) ∈ H v × P ′ , \rho_v(h,p)=p\qquad(h,p)\in H_v\times P', ρ v ( h , p ) = p ( h , p ) ∈ H v × P ′ , and
ρ v ( g , r ) = r ( g , r ) ∈ G v × R v . \rho_v(g,r)=r\qquad(g,r)\in G_v\times R_v. ρ v ( g , r ) = r ( g , r ) ∈ G v × R v . The canonical inclusion P ′ ↪ P Γ P'\hookrightarrow P_\Gamma P ′ ↪ P Γ ρ v \rho_v ρ v
Proposition 7.8 (Kernel of pair deletion). Let K v = ker ( ρ v ) K_v=\ker(\rho_v) K v = ker ( ρ v )
K v ≅ ∗ H v P ′ / R v G v . K_v\cong\ast_{H_v}^{P'/R_v}G_v. K v ≅ ∗ H v P ′ / R v G v . In other words, K v K_v K v G v G_v G v P ′ / R v P'/R_v P ′ / R v H v H_v H v
Proof. Let T T T
P Γ = ( H v × P ′ ) ∗ H v × R v ( G v × R v ) . P_\Gamma=(H_v\times P')\ast_{H_v\times R_v}(G_v\times R_v). P Γ = ( H v × P ′ ) ∗ H v × R v ( G v × R v ) . Since K v K_v K v P Γ / K v ≅ P ′ P_\Gamma/K_v\cong P' P Γ / K v ≅ P ′ K v \ T K_v\backslash T K v \ T H v × P ′ H_v\times P' H v × P ′ G v × R v G_v\times R_v G v × R v P ′ / R v P'/R_v P ′ / R v
The corresponding stabilizers are K v ∩ ( H v × P ′ ) = H v K_v\cap(H_v\times P')=H_v K v ∩ ( H v × P ′ ) = H v K v ∩ ( G v × R v ) = G v K_v\cap(G_v\times R_v)=G_v K v ∩ ( G v × R v ) = G v K v ∩ ( H v × R v ) = H v K_v\cap(H_v\times R_v)=H_v K v ∩ ( H v × R v ) = H v H v H_v H v P ′ P' P ′ H v H_v H v
K v ≅ ∗ H v P ′ / R v G v . K_v\cong\ast_{H_v}^{P'/R_v}G_v. K v ≅ ∗ H v P ′ / R v G v . Proposition 7.9 (Cyclic preimages). Let C ≤ P ′ C \leq P' C ≤ P ′ ρ v − 1 ( C ) ≅ ( ∗ P ′ / R v H v G v ) ⋊ C \rho_v^{-1}(C) \cong\left(\mathop{\ast}_{P'/R_v}^{H_v} G_v\right) \rtimes C ρ v − 1 ( C ) ≅ ( ∗ P ′ / R v H v G v ) ⋊ C C C C P ′ / R v P'/R_v P ′ / R v C C C H v H_v H v
Proof. Since ρ v \rho_v ρ v P ′ ↪ P Γ P' \hookrightarrow P_{\Gamma} P ′ ↪ P Γ P Γ ≅ K v ⋊ P ′ P_{\Gamma} \cong K_v \rtimes P' P Γ ≅ K v ⋊ P ′ C ≤ P ′ C \leq P' C ≤ P ′ ρ v − 1 ( C ) ≅ K v ⋊ C \rho_v^{-1}(C) \cong K_v \rtimes C ρ v − 1 ( C ) ≅ K v ⋊ C K v ≅ ∗ P ′ / R v H v G v K_v \cong\mathop{\ast}_{P'/R_v}^{H_v} G_v K v ≅ ∗ P ′ / R v H v G v C C C C C C P ′ / R v P'/R_v P ′ / R v H v H_v H v P ′ P' P ′ C C C H v H_v H v
7.2.2. FJ-admissible pairs. The preceding propositions reduce the Farrell–Jones problem for graph products of pairs to a local combination problem. For a nonempty set S S S ∗ s ∈ S H G \mathop{\ast}_{s\in S}^{H}G ∗ s ∈ S H G G s ≅ G G_s \cong G G s ≅ G S S S H H H S S S Z \mathbb{Z} Z Z \mathbb{Z} Z S S S H H H G ( G , H ; S ) = ( ∗ s ∈ S H G ) ⋊ Z \mathcal{G}(G,H;S)=\left(\mathop{\ast}_{s\in S}^{H}G\right)\rtimes\mathbb{Z} G ( G , H ; S ) = ( ∗ s ∈ S H G ) ⋊ Z
Definition 7.10. A pair H ≤ G H \leq G H ≤ G FJ-admissible if G ∈ C F J G \in\mathrm{C}_{\mathrm{FJ}} G ∈ C FJ G ( G , H ; S ) ∈ C F J \mathcal{G}(G,H;S) \in\mathrm{C}_{\mathrm{FJ}} G ( G , H ; S ) ∈ C FJ Z \mathbb{Z} Z S S S
For a nonempty Z \mathbb{Z} Z S S S G ( G , H ; S ) = ( ∗ s ∈ S H G ) ⋊ Z \mathcal{G}(G,H;S)=\left(\mathop{\ast}_{s\in S}^{H}G\right)\rtimes\mathbb{Z} G ( G , H ; S ) = ( ∗ s ∈ S H G ) ⋊ Z S S S Z \mathbb{Z} Z H × Z H \times\mathbb{Z} H × Z
Lemma 7.11. Let S = S 1 ⊔ S 2 S=S_1\sqcup S_2 S = S 1 ⊔ S 2 Z \mathbb{Z} Z
G ( G , H ; S ) ≅ G ( G , H ; S 1 ) ∗ H × Z G ( G , H ; S 2 ) . \mathcal{G}(G,H;S)\cong\mathcal{G}(G,H;S_1)\mathop{\ast}_{H\times\mathbb{Z}}\mathcal{G}(G,H;S_2). G ( G , H ; S ) ≅ G ( G , H ; S 1 ) ∗ H × Z G ( G , H ; S 2 ) . Moreover:
if S ≅ Z / m Z S\cong\mathbb{Z}/m\mathbb{Z} S ≅ Z / m Z G ( G , H ; S ) \mathcal{G}(G,H;S) G ( G , H ; S ) ( ∗ s ∈ Z / m Z H G ) × Z \left(\mathop{\ast}_{s\in\mathbb{Z}/m\mathbb{Z}}^{H}G\right)\times\mathbb{Z} ( ∗ s ∈ Z / m Z H G ) × Z m m m
if S ≅ Z S\cong\mathbb{Z} S ≅ Z G ( G , H ; S ) ≅ G ∗ H ( H × Z ) \mathcal{G}(G,H;S)\cong G\mathop{\ast}_{H}(H\times\mathbb{Z}) G ( G , H ; S ) ≅ G ∗ H ( H × Z )
Proof. Write K ( S ) = ∗ s ∈ S H G K(S)=\mathop{\ast}_{s\in S}^{H}G K ( S ) = ∗ s ∈ S H G S = S 1 ⊔ S 2 S=S_1\sqcup S_2 S = S 1 ⊔ S 2 Z \mathbb{Z} Z K ( S ) ≅ K ( S 1 ) ∗ H K ( S 2 ) K(S)\cong K(S_1)\mathop{\ast}_{H}K(S_2) K ( S ) ≅ K ( S 1 ) ∗ H K ( S 2 ) t t t Z \mathbb{Z} Z H H H
K ( S ) ⋊ ⟨ t ⟩ ≅ ( K ( S 1 ) ⋊ ⟨ t ⟩ ) ∗ H × ⟨ t ⟩ ( K ( S 2 ) ⋊ ⟨ t ⟩ ) , K(S)\rtimes\langle t\rangle\cong\left(K(S_1)\rtimes\langle t\rangle\right)\mathop{\ast}_{H\times\langle t\rangle}\left(K(S_2)\rtimes\langle t\rangle\right), K ( S ) ⋊ ⟨ t ⟩ ≅ ( K ( S 1 ) ⋊ ⟨ t ⟩ ) ∗ H × ⟨ t ⟩ ( K ( S 2 ) ⋊ ⟨ t ⟩ ) , which proves
G ( G , H ; S ) ≅ G ( G , H ; S 1 ) ∗ H × Z G ( G , H ; S 2 ) . \mathcal{G}(G,H;S)\cong\mathcal{G}(G,H;S_1)\mathop{\ast}_{H\times\mathbb{Z}}\mathcal{G}(G,H;S_2). G ( G , H ; S ) ≅ G ( G , H ; S 1 ) ∗ H × Z G ( G , H ; S 2 ) . Now suppose that S ≅ Z / m Z S\cong\mathbb{Z}/m\mathbb{Z} S ≅ Z / m Z t m t^m t m G s G_s G s
( ∗ s ∈ Z / m Z H G ) ⋊ m Z ≅ ( ∗ s ∈ Z / m Z H G ) × Z . \left(\mathop{\ast}_{s\in\mathbb{Z}/m\mathbb{Z}}^{H}G\right)\rtimes m\mathbb{Z}\cong\left(\mathop{\ast}_{s\in\mathbb{Z}/m\mathbb{Z}}^{H}G\right)\times\mathbb{Z}. ( ∗ s ∈ Z / m Z H G ) ⋊ m Z ≅ ( ∗ s ∈ Z / m Z H G ) × Z . This subgroup has index m m m G ( G , H ; S ) \mathcal{G}(G,H;S) G ( G , H ; S )
Finally, suppose that S ≅ Z S\cong\mathbb{Z} S ≅ Z 0 0 0 t t t
G ( G , H ; Z ) = ⟨ G , t ∣ t h t − 1 = h , h ∈ H ⟩ ≅ G ∗ H ( H × Z ) . \mathcal{G}(G,H;\mathbb{Z})=\langle G,t\mid tht^{-1}=h,\ h\in H\rangle\cong G\mathop{\ast}_{H}(H\times\mathbb{Z}). G ( G , H ; Z ) = ⟨ G , t ∣ t h t − 1 = h , h ∈ H ⟩ ≅ G ∗ H ( H × Z ) . □
We now use Knopf’s combination theorem for groups acting acylindrically on trees [18 ] . We first fix the convention on acylindricity.
Definition 7.12. Let k ≥ 0 k \ge0 k ≥ 0 G G G T T T k k k k k k acylindrical if it is k k k k k k
This is the convention used by Delzant [10 Definition, p. 1215] and Knopf [18 Definition 3.1] ; it differs slightly from Sela’s original indexing convention [20 ] and from Bowditch’s metric formulation [5 ] .
We shall use the following consequence of Knopf’s theorem.
Theorem 7.13 (Knopf [18 Corollary 4.4] ). Let G G G T T T G ↷ T G \curvearrowright T G ↷ T C F J \mathrm{CFJ} CFJ G ∈ C F J G \in\mathrm{CFJ} G ∈ CFJ
Theorem 7.13 provides the main geometric tool for establishing FJ-admissibility. The difficulty is that the Z \mathbb{Z} Z H × Z H \times\mathbb{Z} H × Z
Theorem 7.14 (Local-to-global Farrell–Jones theorem). Let Γ \Gamma Γ { ( G v , H v ) } v ∈ V ( Γ ) \{(G_v,H_v)\}_{v \in V(\Gamma)} {( G v , H v ) } v ∈ V ( Γ ) Π Γ ( G v , H v ) ∈ C F J \Pi_\Gamma(G_v,H_v) \in\mathrm{CFJ} Π Γ ( G v , H v ) ∈ CFJ
Proof. We argue by induction on ∣ V ( Γ ) ∣ |V(\Gamma)| ∣ V ( Γ ) ∣ Γ \Gamma Γ v v v Π Γ ( G v , H v ) ≅ G v \Pi_\Gamma(G_v,H_v) \cong G_v Π Γ ( G v , H v ) ≅ G v G v ∈ C F J G_v \in\mathrm{CFJ} G v ∈ CFJ v ∈ V ( Γ ) v \in V(\Gamma) v ∈ V ( Γ )
P = Π Γ ( G i , H i ) , P ′ = Π Γ − { v } ( G i , H i ) . P = \Pi_\Gamma(G_i,H_i), \qquad P' = \Pi_{\Gamma-\{v\}}(G_i,H_i). P = Π Γ ( G i , H i ) , P ′ = Π Γ − { v } ( G i , H i ) . By the induction hypothesis, P ′ ∈ C F J P' \in\mathrm{CFJ} P ′ ∈ CFJ ρ v : P ⟶ P ′ \rho_v:P \longrightarrow P' ρ v : P ⟶ P ′ K v : = ker ( ρ v ) ≅ ∗ H v P ′ / R v G v K_v := \ker(\rho_v) \cong\mathop{*}_{H_v}^{P'/R_v} G_v K v := ker ( ρ v ) ≅ ∗ H v P ′ / R v G v P ′ / R v P'/R_v P ′ / R v Z \mathbb{Z} Z ( G v , H v ) (G_v,H_v) ( G v , H v ) G ( G v , H v ; P ′ / R v ) ≅ K v × Z \mathcal{G}(G_v,H_v;P'/R_v) \cong K_v \times\mathbb{Z} G ( G v , H v ; P ′ / R v ) ≅ K v × Z C F J \mathrm{CFJ} CFJ K v ∈ C F J K_v \in\mathrm{CFJ} K v ∈ CFJ C ≤ P ′ C \le P' C ≤ P ′
ρ v − 1 ( C ) ≅ ( ∗ H v P ′ / R v G v ) ⋊ C . \rho_v^{-1}(C) \cong\left(\mathop{*}_{H_v}^{P'/R_v} G_v\right) \rtimes C. ρ v − 1 ( C ) ≅ ( ∗ H v P ′ / R v G v ) ⋊ C . Since C ≅ Z C \cong\mathbb{Z} C ≅ Z P ′ / R v P'/R_v P ′ / R v H v H_v H v ( G v , H v ) (G_v,H_v) ( G v , H v ) ρ v − 1 ( C ) ∈ C F J \rho_v^{-1}(C) \in\mathrm{CFJ} ρ v − 1 ( C ) ∈ CFJ P ∈ C F J P \in\mathrm{CFJ} P ∈ CFJ
Corollary 7.15. Let Z = Z L ( A , B ) Z = Z_L(A,B) Z = Z L ( A , B ) G i ↷ E i G_i \curvearrowright E_i G i ↷ E i H i ≤ G i H_i \le G_i H i ≤ G i E i ≅ G i / H i E_i \cong G_i/H_i E i ≅ G i / H i ( G i , H i ) (G_i,H_i) ( G i , H i ) π 1 ( Z ) ≅ Π L ( G i , H i ) \pi_1(Z) \cong\Pi_{L}(G_i,H_i) π 1 ( Z ) ≅ Π L ( G i , H i )
Proof. The fundamental-group description gives π 1 ( Z ) ≅ Π L ( G i , H i ) \pi_1(Z) \cong\Pi_L(G_i,H_i) π 1 ( Z ) ≅ Π L ( G i , H i )
Concrete classes and alternative criteria
Theorem 7.14 reduces the Farrell–Jones conjecture for graph products of pairs to the vertexwise FJ-admissibility condition. We now discuss several situations in which the required Farrell–Jones conclusion follows from existing results. These cases arise from different mechanisms: the Gandini–Rüping theorem for ordinary graph products, combination theorems for acylindrical actions on trees, extension arguments for normal pairs, and the Coxeter-group description of finite group pairs.
Ordinary graph products.
The first case is the ordinary graph-product situation. If H = 1 H=1 H = 1 ∗ H G = ∗ s ∈ S G *_{H}G=*_{s\in S}G ∗ H G = ∗ s ∈ S G G ( G , 1 ; S ) = ( ∗ s ∈ S G ) ⋊ Z \mathcal{G}(G,1;S)=\left(*_{s\in S}G\right)\rtimes\mathbb{Z} G ( G , 1 ; S ) = ( ∗ s ∈ S G ) ⋊ Z Z \mathbb{Z} Z S S S
Gandini–Rüping prove precisely that this group satisfies the Farrell–Jones conjecture whenever G G G
Proposition 7.17. If G ∈ C F J G\in\mathcal{C}_{\mathrm{FJ}} G ∈ C FJ ( G , 1 ) (G,1) ( G , 1 ) F J FJ F J
Consequently, Theorem 7.14 contains the Gandini–Rüping theorem for ordinary graph products as the special case in which all H i H_i H i
Acylindrical pairs.
The next source of FJ-admissible pairs comes from Bass–Serre theory. Let H ≤ G H\leq G H ≤ G G ∈ C F J G\in\mathcal{C}_{\mathrm{FJ}} G ∈ C FJ Z \mathbb{Z} Z O \mathcal{O} O O ≅ Z / m Z \mathcal{O}\cong\mathbb{Z}/m\mathbb{Z} O ≅ Z / m Z G ( G , H ; O ) \mathcal{G}(G,H;\mathcal{O}) G ( G , H ; O ) ( ∗ H G ) × Z \left(*_{H}G\right)\times\mathbb{Z} ( ∗ H G ) × Z m m m O ≅ Z \mathcal{O}\cong\mathbb{Z} O ≅ Z G ( G , H ; Z ) ≅ G ∗ H ( H × Z ) \mathcal{G}(G,H;\mathbb{Z})\cong G*_{H}(H\times\mathbb{Z}) G ( G , H ; Z ) ≅ G ∗ H ( H × Z )
Proposition 7.18 (Single-orbit acylindrical criterion). Let H ≤ G H\leq G H ≤ G G ∈ C F J G\in\mathcal{C}_{\mathrm{FJ}} G ∈ C FJ O \mathcal{O} O Z \mathbb{Z} Z
(1) If O ≅ Z / m Z \mathcal{O}\cong\mathbb{Z}/m\mathbb{Z} O ≅ Z / m Z ∗ H G *_{H}G ∗ H G G ( G , H ; O ) ∈ C F J \mathcal{G}(G,H;\mathcal{O})\in\mathcal{C}_{\mathrm{FJ}} G ( G , H ; O ) ∈ C FJ
(2) If O ≅ Z \mathcal{O}\cong\mathbb{Z} O ≅ Z G ∗ H ( H × Z ) G*_{H}(H\times\mathbb{Z}) G ∗ H ( H × Z ) G ( G , H ; O ) ∈ C F J \mathcal{G}(G,H;\mathcal{O})\in\mathcal{C}_{\mathrm{FJ}} G ( G , H ; O ) ∈ C FJ
Proof. For (1), the Bass–Serre tree of ∗ H G *_{H}G ∗ H G G G G H H H C F J \mathcal{C}_{\mathrm{FJ}} C FJ G G G H H H ∗ H G ∈ C F J *_{H}G\in\mathcal{C}_{\mathrm{FJ}} ∗ H G ∈ C FJ C F J \mathcal{C}_{\mathrm{FJ}} C FJ ( ∗ H G ) × Z ∈ C F J \left(*_{H}G\right)\times\mathbb{Z}\in\mathcal{C}_{\mathrm{FJ}} ( ∗ H G ) × Z ∈ C FJ G ( G , H ; O ) \mathcal{G}(G,H;\mathcal{O}) G ( G , H ; O )
For (2), since H ≤ G H\leq G H ≤ G H ∈ C F J H\in\mathcal{C}_{\mathrm{FJ}} H ∈ C FJ H × Z ∈ C F J H\times\mathbb{Z}\in\mathcal{C}_{\mathrm{FJ}} H × Z ∈ C FJ G ( G , H ; Z ) ≅ G ∗ H ( H × Z ) \mathcal{G}(G,H;\mathbb{Z})\cong G*_{H}(H\times\mathbb{Z}) G ( G , H ; Z ) ≅ G ∗ H ( H × Z )
For several orbits an additional difficulty appears. If S = S 1 ⊔ S 2 S=S_1\sqcup S_2 S = S 1 ⊔ S 2 G ( G , H ; S ) ≅ G ( G , H ; S 1 ) ∗ H × Z G ( G , H ; S 2 ) \mathcal{G}(G,H;S)\cong\mathcal{G}(G,H;S_1)*_{H\times\mathbb{Z}}\mathcal{G}(G,H;S_2) G ( G , H ; S ) ≅ G ( G , H ; S 1 ) ∗ H × Z G ( G , H ; S 2 ) Z \mathbb{Z} Z H × Z H\times\mathbb{Z} H × Z
One convenient sufficient condition is the following.
Proposition 7.19 (Acylindrical FJ-admissibility criterion). Let H ≤ G H\leq G H ≤ G G ∈ C F J G\in\mathcal{C}_{\mathrm{FJ}} G ∈ C FJ
(1) for every transitive Z \mathbb{Z} Z O \mathcal{O} O G ( G , H ; O ) \mathcal{G}(G,H;\mathcal{O}) G ( G , H ; O ) C F J \mathcal{C}_{\mathrm{FJ}} C FJ
(2) whenever S 1 S_1 S 1 S 2 S_2 S 2 Z \mathbb{Z} Z G ( G , H ; S 1 ) ∗ H × Z G ( G , H ; S 2 ) \mathcal{G}(G,H;S_1)*_{H\times\mathbb{Z}}\mathcal{G}(G,H;S_2) G ( G , H ; S 1 ) ∗ H × Z G ( G , H ; S 2 )
Then ( G , H ) (G,H) ( G , H )
Proof. By Lemma 7.11 and Theorem 7.13 , induction on the number of orbits shows that G ( G , H ; S ) ∈ C F J \mathcal{G}(G,H;S)\in\mathcal{C}_{\mathrm{FJ}} G ( G , H ; S ) ∈ C FJ S S S Z \mathbb{Z} Z
For an arbitrary Z \mathbb{Z} Z S S S S \mathcal{S} S S S S G ( G , H ; S ) ≅ c o l i m S 0 ∈ S G ( G , H ; S 0 ) \mathcal{G}(G,H;S)\cong\mathop{\mathrm{colim}}_{S_0\in\mathcal{S}}\mathcal{G}(G,H;S_0) G ( G , H ; S ) ≅ colim S 0 ∈ S G ( G , H ; S 0 ) C F J \mathcal{C}_{\mathrm{FJ}} C FJ G ( G , H ; S ) ∈ C F J \mathcal{G}(G,H;S)\in\mathcal{C}_{\mathrm{FJ}} G ( G , H ; S ) ∈ C FJ ( G , H ) (G,H) ( G , H )
Remark 7.20. The preceding proposition is deliberately formulated as a criterion rather than as a classification of FJ-admissible pairs. In concrete applications one may verify acylindricity through almost malnormality or other geometric hypotheses on the inclusion H ≤ G H \leq G H ≤ G H ≤ G H \leq G H ≤ G almost malnormal if H ∩ g H g − 1 H \cap gHg^{-1} H ∩ g H g − 1 g ∈ G ∖ H g \in G \setminus H g ∈ G ∖ H [18 Definition 3.3] ). Determining the largest natural class of pairs for which these conditions hold remains an interesting problem.
Free-factor pairs.
There is also a non-normal class which can be treated directly using the free-product combination results of Gandini–Rüping [GR13].
Proposition 7.21 (Free-factor pairs). Suppose that G = H ∗ K G = H \ast K G = H ∗ K H , K ∈ C F J H, K \in\mathrm{C}_{\mathrm{FJ}} H , K ∈ C FJ ( G , H ) (G, H) ( G , H )
Proof. Since G = H ∗ K G = H \ast K G = H ∗ K H , K ∈ C F J H, K \in\mathrm{C}_{\mathrm{FJ}} H , K ∈ C FJ G ∈ C F J G \in\mathrm{C}_{\mathrm{FJ}} G ∈ C FJ S S S Z \mathbb{Z} Z G = H ∗ K G = H \ast K G = H ∗ K G G G H H H
s ∈ S ∗ H G ≅ H ∗ ( s ∈ S ∗ K ) . {}_{s \in S}^{\ast} H G \cong H \ast\left({}_{s \in S}^{\ast} K\right). s ∈ S ∗ H G ≅ H ∗ ( s ∈ S ∗ K ) . Under this identification, the Z \mathbb{Z} Z H H H K K K S S S K S = s ∈ S ∗ K K_S = {}_{s \in S}^{\ast} K K S = s ∈ S ∗ K K S ⋊ Z ∈ C F J K_S \rtimes\mathbb{Z} \in\mathrm{C}_{\mathrm{FJ}} K S ⋊ Z ∈ C FJ H ⋊ i d Z = H × Z ∈ C F J H \rtimes_{\mathrm{id}} \mathbb{Z} = H \times\mathbb{Z} \in\mathrm{C}_{\mathrm{FJ}} H ⋊ id Z = H × Z ∈ C FJ H ∗ K S H \ast K_S H ∗ K S H H H K S K_S K S ( H ∗ K S ) ⋊ Z ∈ C F J (H \ast K_S) \rtimes\mathbb{Z} \in\mathrm{C}_{\mathrm{FJ}} ( H ∗ K S ) ⋊ Z ∈ C FJ G ( G , H ; S ) ∈ C F J \mathcal{G}(G, H; S) \in\mathrm{C}_{\mathrm{FJ}} G ( G , H ; S ) ∈ C FJ S S S ( G , H ) (G, H) ( G , H )
Example 7.22. Let G = F ( a , b ) = ⟨ a ⟩ ∗ ⟨ b ⟩ G = F(a,b) = \langle a\rangle\ast\langle b\rangle G = F ( a , b ) = ⟨ a ⟩ ∗ ⟨ b ⟩ H = ⟨ a ⟩ H = \langle a\rangle H = ⟨ a ⟩ ⟨ a ⟩ ≅ Z \langle a\rangle\cong\mathbb{Z} ⟨ a ⟩ ≅ Z ⟨ b ⟩ ≅ Z \langle b\rangle\cong\mathbb{Z} ⟨ b ⟩ ≅ Z C F J \mathrm{C}_{\mathrm{FJ}} C FJ ( F 2 , ⟨ a ⟩ ) (F_2,\langle a\rangle) ( F 2 , ⟨ a ⟩)
Normal pairs and cyclic extensions.
The normal case admits a stronger formulation which isolates the precise local property needed in the pair-deletion argument. Namely, what is required is not virtual solvability, or even amenability itself, but stability of the Farrell–Jones conjecture under cyclic extensions of the common subgroup.
Definition 7.23. A group H H H FJ-stable under cyclic extensions if H ∈ C F J H \in\mathrm{C}_{\mathrm{FJ}} H ∈ C FJ α ∈ Aut ( H ) \alpha\in\operatorname{Aut}(H) α ∈ Aut ( H ) H ⋊ α Z H \rtimes_{\alpha} \mathbb{Z} H ⋊ α Z C F J \mathrm{C}_{\mathrm{FJ}} C FJ
Remark 7.24. The condition in Definition 7.23 is tailored to the Farrell–Jones inheritance principle. Indeed, whenever
1 ⟶ H ⟶ E ⟶ Z ⟶ 1 1 \longrightarrow H \longrightarrow E \longrightarrow\mathbb{Z} \longrightarrow1 1 ⟶ H ⟶ E ⟶ Z ⟶ 1 is exact, the extension splits and E ≅ H ⋊ α Z E \cong H \rtimes_{\alpha} \mathbb{Z} E ≅ H ⋊ α Z α ∈ Aut ( H ) \alpha\in\operatorname{Aut}(H) α ∈ Aut ( H ) H H H
Remark 7.25. The appearance of cyclic extensions is not accidental. Wu’s proof of the Farrell–Jones conjecture for elementary amenable groups of finite Hirsch length likewise reduces, through the inheritance principle, to groups which are extensions by an infinite cyclic group. This is precisely the type of subgroup produced by the cyclic preimages in the pair-deletion argument.
Proposition 7.26 (Normal FJ-admissible pairs). Let H ≤ G H \leq G H ≤ G Q = G / H Q = G/H Q = G / H Q ∈ C F J Q \in\mathrm{C}_{\mathrm{FJ}} Q ∈ C FJ H H H ( G , H ) (G,H) ( G , H )
Proof. We first show that G ∈ C F J G \in\mathrm{C}_{\mathrm{FJ}} G ∈ C FJ q : G ⟶ Q q\colon G \longrightarrow Q q : G ⟶ Q Q ∈ C F J Q \in\mathrm{C}_{\mathrm{FJ}} Q ∈ C FJ q q q H H H C F J \mathrm{C}_{\mathrm{FJ}} C FJ H H H C ≤ Q C \leq Q C ≤ Q
1 ⟶ H ⟶ q − 1 ( C ) ⟶ C ⟶ 1 1 \longrightarrow H \longrightarrow q^{-1}(C) \longrightarrow C \longrightarrow1 1 ⟶ H ⟶ q − 1 ( C ) ⟶ C ⟶ 1 is exact. Since C ≅ Z C \cong\mathbb{Z} C ≅ Z q ˉ − 1 ( C ) ≅ H ⋊ α Z \bar{q}^{-1}(C) \cong H \rtimes_{\alpha} \mathbb{Z} q ˉ − 1 ( C ) ≅ H ⋊ α Z α ∈ Aut ( H ) \alpha\in\operatorname{Aut}(H) α ∈ Aut ( H ) q ˉ − 1 ( C ) ∈ C F J \bar{q}^{-1}(C) \in\mathcal{C}_{\mathrm{FJ}} q ˉ − 1 ( C ) ∈ C FJ G ∈ C F J G \in\mathcal{C}_{\mathrm{FJ}} G ∈ C FJ
Let S S S Z \mathbb{Z} Z H ⊴ G H \trianglelefteq G H ⊴ G H H H ∗ s ∈ S H G \ast_{s\in S}^{H}G ∗ s ∈ S H G Z \mathbb{Z} Z H H H
1 ⟶ H ⟶ G ( G , H ; S ) → q ˉ ( ∗ s ∈ S Q ) ⋊ Z ⟶ 1 1 \longrightarrow H \longrightarrow\mathcal{G}(G,H;S) \xrightarrow{\bar{q}} \left(\ast_{s\in S} Q\right) \rtimes\mathbb{Z} \longrightarrow1 1 ⟶ H ⟶ G ( G , H ; S ) q ˉ ( ∗ s ∈ S Q ) ⋊ Z ⟶ 1 By [GR13, Lemma 2.8], ( ∗ s ∈ S Q ) ⋊ Z ∈ C F J \left(\ast_{s\in S} Q\right) \rtimes\mathbb{Z} \in\mathcal{C}_{\mathrm{FJ}} ( ∗ s ∈ S Q ) ⋊ Z ∈ C FJ ker ( q ˉ ) = H ∈ C F J \ker(\bar{q})=H \in\mathcal{C}_{\mathrm{FJ}} ker ( q ˉ ) = H ∈ C FJ C C C
1 ⟶ H ⟶ q ˉ − 1 ( C ) ⟶ C ⟶ 1 1 \longrightarrow H \longrightarrow\bar{q}^{-1}(C) \longrightarrow C \longrightarrow1 1 ⟶ H ⟶ q ˉ − 1 ( C ) ⟶ C ⟶ 1 is exact, and hence q ˉ − 1 ( C ) ≅ H ⋊ β Z \bar{q}^{-1}(C) \cong H \rtimes_{\beta} \mathbb{Z} q ˉ − 1 ( C ) ≅ H ⋊ β Z β ∈ Aut ( H ) \beta\in\operatorname{Aut}(H) β ∈ Aut ( H ) H H H q ˉ − 1 ( C ) ∈ C F J \bar{q}^{-1}(C) \in\mathcal{C}_{\mathrm{FJ}} q ˉ − 1 ( C ) ∈ C FJ G ( G , H ; S ) ∈ C F J \mathcal{G}(G,H;S) \in\mathcal{C}_{\mathrm{FJ}} G ( G , H ; S ) ∈ C FJ Z \mathbb{Z} Z S S S ( G , H ) (G,H) ( G , H )
Proposition 7.27 (Normal pairs with Farrell–Jones quotient). Let H ⊴ G H \trianglelefteq G H ⊴ G Q = G / H Q=G/H Q = G / H G , Q ∈ C F J G,Q \in\mathcal{C}_{\mathrm{FJ}} G , Q ∈ C FJ ( G , H ) (G,H) ( G , H )
Proof. Let S S S Z \mathbb{Z} Z H ⊴ G H \trianglelefteq G H ⊴ G
1 ⟶ H ⟶ G ( G , H ; S ) → q ˉ ( ∗ s ∈ S Q ) ⋊ Z ⟶ 1 1 \longrightarrow H \longrightarrow\mathcal{G}(G,H;S) \xrightarrow{\bar{q}} \left(\ast_{s\in S} Q\right) \rtimes\mathbb{Z} \longrightarrow1 1 ⟶ H ⟶ G ( G , H ; S ) q ˉ ( ∗ s ∈ S Q ) ⋊ Z ⟶ 1 By [GR13, Lemma 2.8], ( ∗ s ∈ S Q ) ⋊ Z ∈ C F J \left(\ast_{s\in S} Q\right) \rtimes\mathbb{Z} \in\mathcal{C}_{\mathrm{FJ}} ( ∗ s ∈ S Q ) ⋊ Z ∈ C FJ H ∈ C F J H \in\mathcal{C}_{\mathrm{FJ}} H ∈ C FJ H ≤ G H \leq G H ≤ G C C C c c c C C C t = k τ n ∈ G ( G , H ; S ) t=k\tau^{n}\in\mathcal{G}(G,H;S) t = k τ n ∈ G ( G , H ; S ) k ∈ ∗ s ∈ S H G k\in\ast_{s\in S}^{H}G k ∈ ∗ s ∈ S H G τ \tau τ Z \mathbb{Z} Z fold : ∗ s ∈ S H G ⟶ G \operatorname{fold}:\ast_{s\in S}^{H}G\longrightarrow G fold : ∗ s ∈ S H G ⟶ G G G G g = fold ( k ) g=\operatorname{fold}(k) g = fold ( k ) τ \tau τ H H H t t t H H H g g g q ˉ − 1 ( C ) ≅ H ⋊ c g Z \bar{q}^{-1}(C)\cong H\rtimes_{c_g}\mathbb{Z} q ˉ − 1 ( C ) ≅ H ⋊ c g Z
H ⋊ c g Z ⟶ G × Z , ( h , m ) ⟼ ( h g m , m ) , H\rtimes_{c_g}\mathbb{Z}\longrightarrow G\times\mathbb{Z},\qquad(h,m)\longmapsto(hg^{m},m), H ⋊ c g Z ⟶ G × Z , ( h , m ) ⟼ ( h g m , m ) , is injective. Hence q ˉ − 1 ( C ) ≤ G × Z \bar{q}^{-1}(C)\leq G\times\mathbb{Z} q ˉ − 1 ( C ) ≤ G × Z G × Z ∈ C F J G\times\mathbb{Z}\in\mathcal{C}_{\mathrm{FJ}} G × Z ∈ C FJ q ˉ − 1 ( C ) ∈ C F J \bar{q}^{-1}(C)\in\mathcal{C}_{\mathrm{FJ}} q ˉ − 1 ( C ) ∈ C FJ G ( G , H ; S ) ∈ C F J \mathcal{G}(G,H;S)\in\mathcal{C}_{\mathrm{FJ}} G ( G , H ; S ) ∈ C FJ Z \mathbb{Z} Z S S S ( G , H ) (G,H) ( G , H )
Remark 7.28. Propositions 7.26 and 7.27 are complementary. The former does not assume G ∈ C F J G\in\mathcal{C}_{\mathrm{FJ}} G ∈ C FJ H H H G ∈ C F J G\in\mathcal{C}_{\mathrm{FJ}} G ∈ C FJ H H H
Corollary 7.29 (Normal virtually solvable pairs). Let H ⊴ G H\trianglelefteq G H ⊴ G Q = G / H Q=G/H Q = G / H H H H Q ∈ C F J Q\in\mathcal{C}_{\mathrm{FJ}} Q ∈ C FJ ( G , H ) (G,H) ( G , H )
Proof. By Wegner’s theorem [21 ] , H ∈ C F J H\in\mathcal{C}_{\mathrm{FJ}} H ∈ C FJ α ∈ Aut ( H ) \alpha\in\operatorname{Aut}(H) α ∈ Aut ( H ) H ⋊ α Z H\rtimes_{\alpha}\mathbb{Z} H ⋊ α Z C F J \mathcal{C}_{\mathrm{FJ}} C FJ H H H
Corollary 7.30 (Normal elementary amenable pairs). Let H ⊴ G H\trianglelefteq G H ⊴ G Q = G / H Q=G/H Q = G / H H H H Q ∈ C F J Q\in\mathcal{C}_{\mathrm{FJ}} Q ∈ C FJ ( G , H ) (G,H) ( G , H )
Proof. By [22 Theorem A] , the group H H H α ∈ Aut ( H ) \alpha\in\operatorname{Aut}(H) α ∈ Aut ( H ) H ⋊ α Z H\rtimes_{\alpha}\mathbb{Z} H ⋊ α Z h ( H ⋊ α Z ) ≤ h ( H ) + 1 < ∞ h(H\rtimes_{\alpha}\mathbb{Z})\leq h(H)+1<\infty h ( H ⋊ α Z ) ≤ h ( H ) + 1 < ∞ H ⋊ α Z ∈ C F J H\rtimes_{\alpha}\mathbb{Z}\in\mathcal{C}_{\mathrm{FJ}} H ⋊ α Z ∈ C FJ [22 Theorem A] . Thus H H H
Remark 7.31 (Houghton-type examples). Corollary 7.30 is genuinely stronger than the corresponding virtually solvable statement. Wu [22 ] points out that Houghton groups give elementary amenable groups of finite Hirsch length which are not virtually solvable. For example, let H H H Q ∈ C F J Q \in\mathcal{C}_{\mathrm{FJ}} Q ∈ C FJ G = H × Q G = H \times Q G = H × Q H ⊴ G H \trianglelefteq G H ⊴ G G / H ≅ Q G/H \cong Q G / H ≅ Q ( G , H ) (G,H) ( G , H )
An alternative global normal-pair criterion. When all subgroups H i H_i H i H i ⊴ G i H_i \trianglelefteq G_i H i ⊴ G i i i i Q i = G i / H i Q_i = G_i/H_i Q i = G i / H i
1 ⟶ N ⟶ Π L 1 ( G i , H i ) → ρ Q ⟶ 1 , 1 \longrightarrow N \longrightarrow\Pi_{L_1}(G_i,H_i) \xrightarrow{\rho} Q \longrightarrow1, 1 ⟶ N ⟶ Π L 1 ( G i , H i ) ρ Q ⟶ 1 , where
N = ∏ i H i , Q = ∗ L 1 Q i . N = \prod_i H_i,\qquad Q = \ast_{L_1} Q_i. N = i ∏ H i , Q = ∗ L 1 Q i . Proposition 7.32 (Global normal-pair criterion). Assume that:
(1) every Q i Q_i Q i C F J \mathcal{C}_{\mathrm{FJ}} C FJ
(2) N = ∏ i H i N = \prod_i H_i N = ∏ i H i C F J \mathcal{C}_{\mathrm{FJ}} C FJ
(3) for every infinite cyclic subgroup C ≤ Q C \leq Q C ≤ Q ρ − 1 ( C ) ∈ C F J \rho^{-1}(C) \in\mathcal{C}_{\mathrm{FJ}} ρ − 1 ( C ) ∈ C FJ
Then Π L 1 ( G i , H i ) ∈ C F J \Pi_{L_1}(G_i,H_i) \in\mathcal{C}_{\mathrm{FJ}} Π L 1 ( G i , H i ) ∈ C FJ
Proof. By Theorem 7.3, Q = ∗ L 1 Q i ∈ C F J Q = \ast_{L_1} Q_i \in\mathcal{C}_{\mathrm{FJ}} Q = ∗ L 1 Q i ∈ C FJ ρ \rho ρ N N N C F J \mathcal{C}_{\mathrm{FJ}} C FJ ρ − 1 ( C ) \rho^{-1}(C) ρ − 1 ( C ) C F J \mathcal{C}_{\mathrm{FJ}} C FJ C ≤ Q C \leq Q C ≤ Q
Remark 7.33 (Amenability and the Farrell–Jones problem.). The role of amenability in the present section should be distinguished from that of acylindricity. The latter provides a geometric combination mechanism through Knopf’s theorem, whereas amenability identifies natural classes of local subgroups which may already satisfy the Farrell–Jones conjecture. This distinction is particularly relevant in view of the amenable-category results obtained earlier in the paper. The same local subgroups H i ≤ G i H_i \leq G_i H i ≤ G i [22 Theorem A] .
Summarizing the preceding results and Theorem 7.14, we obtain the following.
Theorem 7.34 (Farrell–Jones combination theorem for graph products of pairs). Let Γ \Gamma Γ { ( G v , H v ) } v ∈ V ( Γ ) \{(G_v,H_v)\}_{v\in V(\Gamma)} {( G v , H v ) } v ∈ V ( Γ ) v ∈ V ( Γ ) v \in V(\Gamma) v ∈ V ( Γ )
(i) H v = { 1 } H_v = \{1\} H v = { 1 } G v ∈ C F J G_v \in\mathcal{C}_{\mathrm{FJ}} G v ∈ C FJ
(ii) H v ≤ G v H_v \leq G_v H v ≤ G v G v / H v ∈ C F J G_v/H_v \in\mathcal{C}_{\mathrm{FJ}} G v / H v ∈ C FJ G v ∈ C F J G_v \in\mathcal{C}_{\mathrm{FJ}} G v ∈ C FJ H v H_v H v
(iii) G v = H v ∗ K v G_v = H_v \ast K_v G v = H v ∗ K v H v , K v ∈ C F J H_v,K_v \in\mathcal{C}_{\mathrm{FJ}} H v , K v ∈ C FJ
(iv) G v ∈ C F J G_v \in\mathcal{C}_{\mathrm{FJ}} G v ∈ C FJ ( G v , H v ) (G_v,H_v) ( G v , H v )
Then Π Γ ( G v , H v ) ∈ C F J \Pi_\Gamma(G_v,H_v) \in\mathcal{C}_{\mathrm{FJ}} Π Γ ( G v , H v ) ∈ C FJ
Remark 7.35. Condition (ii) of Theorem 7.34 contains two important concrete classes. If H v H_v H v H v H_v H v H v H_v H v H v H_v H v [22 Theorem A] . Thus, provided G v / H v ∈ C F J G_v/H_v \in\mathcal{C}_{\mathrm{FJ}} G v / H v ∈ C FJ
The importance of Theorem 7.34 is that its hypotheses are vertexwise: different vertices may be treated by different mechanisms. Thus a single graph product of pairs may simultaneously contain ordinary graph-product vertices, free-factor pairs, normal pairs treated by cyclic-extension methods, and non-normal pairs controlled by acylindrical Bass–Serre actions.
Corollary 7.36 (Farrell–Jones for polyhedral products). Let Z = Z L ( A , B ) Z=Z_L(A,B) Z = Z L ( A , B ) polyhedral product such that each action G i ↷ E i G_i\curvearrowright E_i G i ↷ E i H i ≤ G i H_i\leq G_i H i ≤ G i E i ≅ G i / H i E_i\cong G_i/H_i E i ≅ G i / H i ( G i , H i ) (G_i,H_i) ( G i , H i ) Theorem 7.34 . Then π 1 ( Z ) ≅ Π L 1 ( G i , H i ) ∈ C F J \pi_1(Z)\cong\Pi_L^1(G_i,H_i)\in\mathrm{CFJ} π 1 ( Z ) ≅ Π L 1 ( G i , H i ) ∈ CFJ
Theorem 7.34 collects the classes for which the local FJ-admissibility condition can be verified directly. There is also an independent global class which does not require this local criterion.
Proposition 7.37. Let G = Π Γ ( G i , H i ) G=\Pi_\Gamma(G_i,H_i) G = Π Γ ( G i , H i ) G i G_i G i G ∈ C F J G\in\mathrm{CFJ} G ∈ CFJ
Proof. Januszkiewicz–Świątkowski [JŚ01, Corollary 5.15] show that graph products of finite group pairs are commensurable with Coxeter groups. Coxeter groups are CAT(0)-groups and hence belong to C F J \mathrm{CFJ} CFJ
The preceding examples show that the Farrell–Jones problem for graph products of pairs admits several complementary approaches. The pair-deletion method is local and applies without normality, provided the vertex pairs are FJ-admissible. Free-factor decompositions give a direct non-normal class through the combination results of Gandini–Rüping, while acylindrical Bass–Serre actions provide a geometric mechanism for treating more general inclusions. For normal pairs, cyclic-extension and quotient arguments give a different route, whereas finite group pairs can be handled globally through their relation with Coxeter groups.
We close this subsection with the following problem.
Question 7.39. Which natural classes of inclusions H ≤ G H\leq G H ≤ G
More specifically, can the acylindricity hypotheses in Proposition 7.19 be replaced by an appropriate relative finite-homotopy F \mathcal{F} F
The preceding results reveal a local-to-global mechanism for rigidity of polyhedral products. The local data are the vertex pairs ( G i , H i ) (G_i,H_i) ( G i , H i ) G i ↷ π 0 ( p i − 1 ( B i ) ) G_i\curvearrowright\pi_0(p_i^{-1}(B_i)) G i ↷ π 0 ( p i − 1 ( B i ))
Thus the Farrell–Jones problem reduces to determining natural classes of FJ-admissible pairs. Any such class immediately yields, through Theorem 7.14 , a Farrell–Jones theorem for the associated graph products of pairs, and hence for the fundamental groups of the corresponding polyhedral products.
For closed aspherical manifold polyhedral products of dimension at least 5, Theorem 7.41 then converts these group-theoretic results into topological rigidity statements.
The Borel conjecture
Recall that a closed manifold M M M topologically rigid if every homotopy equivalence f : N → M f\colon N\to M f : N → M N N N
The connection with the Farrell–Jones conjecture is provided by surgery theory. We shall use the following standard consequence of the work of Bartels and Lück [4 ] .
Theorem 7.40 (Bartels–Lück [4 ] ). Let G G G K K K m ≤ 0 m \le0 m ≤ 0 m = 1 m = 1 m = 1 L L L G G G
In particular, the form of the Farrell–Jones conjecture encoded by C F J \mathrm{CFJ} CFJ
Corollary 7.41 (Borel rigidity for polyhedral products). Let M = Z L ( A , B ) M = \mathcal{Z}_{L}(A,B) M = Z L ( A , B ) π 1 ( M ) ∈ C F J \pi_{1}(M) \in\mathrm{CFJ} π 1 ( M ) ∈ CFJ M M M
Proof. Since M M M π 1 ( M ) \pi_{1}(M) π 1 ( M ) Theorem 7.40 . Hence M M M
Combining this with the Farrell–Jones combination theorem for graph products of pairs gives the following direct application.
Corollary 7.42 (Borel rigidity from graph products of pairs). Let M = Z L ( A , B ) M = \mathcal{Z}_{L}(A,B) M = Z L ( A , B ) G i ↷ E i G_i \curvearrowright E_i G i ↷ E i H i ≤ G i H_i \le G_i H i ≤ G i E i ≅ G i / H i E_i \cong G_i/H_i E i ≅ G i / H i i i i ( G i , H i ) (G_i,H_i) ( G i , H i ) Theorem 7.34 . Then M M M
Proof. By Corollary 7.36, π 1 ( M ) ≅ Π L 1 ( G i , H i ) ∈ C F J \pi_{1}(M) \cong\Pi_{L1}(G_i,H_i) \in\mathrm{CFJ} π 1 ( M ) ≅ Π L 1 ( G i , H i ) ∈ CFJ
We close with several manifold examples illustrating the scope of the preceding results. They show that the local pairs considered above arise naturally from manifold pairs and provide useful consistency checks for the Farrell–Jones and rigidity machinery. The real moment-angle case represents the basic non-transitive situation. Möbius-type polyhedral products give a normal but non-split example, whereas surface-boundary pairs provide natural non-normal inclusions which are not covered by the elementary normal or free-factor criteria.
Example 7.44 (Real moment-angle manifolds cf. [13 ] ). Let L L L Z L R = Z L ( D 1 , S 0 ) \mathcal{Z}_{L}^{\mathbb{R}} = \mathcal{Z}_{L}(D^{1},S^{0}) Z L R = Z L ( D 1 , S 0 ) W L W_{L} W L W L W_{L} W L C A T ( 0 ) \mathrm{CAT}(0) CAT ( 0 ) C F J \mathrm{CFJ} CFJ π 1 ( Z L R ) \pi_{1}(\mathcal{Z}_{L}^{\mathbb{R}}) π 1 ( Z L R ) Z L R \mathcal{Z}_{L}^{\mathbb{R}} Z L R
Notice that the above example lies outside the transitive framework of Corollary 7.42: here G i = 1 G_i = 1 G i = 1 E i = S 0 E_i = S^{0} E i = S 0
Example 7.45 (Möbius-type polyhedral products). Let ( A , B ) = ( M o ¨ , S 1 ) (A,B) = (\mathrm{Mö},S^{1}) ( A , B ) = ( M o ¨ , S 1 ) S 1 = ∂ M o ¨ S^{1} = \partial\mathrm{Mö} S 1 = ∂ M o ¨ π 1 ( M o ¨ ) ≅ Z \pi_{1}(\mathrm{Mö}) \cong\mathbb{Z} π 1 ( M o ¨ ) ≅ Z π 1 ( S 1 ) ≅ 2 Z ≤ Z \pi_{1}(S^{1}) \cong2\mathbb{Z} \le\mathbb{Z} π 1 ( S 1 ) ≅ 2 Z ≤ Z ∂ M o ¨ ↪ M o ¨ \partial\mathrm{Mö} \hookrightarrow\mathrm{Mö} ∂ M o ¨ ↪ M o ¨
π 1 ( ∂ M o ¨ ) ≅ Z ⟶ π 1 ( M o ¨ ) ≅ Z \pi_{1}(\partial\mathrm{Mö}) \cong\mathbb{Z} \longrightarrow\pi_{1}(\mathrm{Mö}) \cong\mathbb{Z} π 1 ( ∂ M o ¨ ) ≅ Z ⟶ π 1 ( M o ¨ ) ≅ Z with image 2 Z 2\mathbb{Z} 2 Z ( Z , 2 Z ) (\mathbb{Z},2\mathbb{Z}) ( Z , 2 Z )
1 ⟶ 2 Z ⟶ Z ⟶ Z / 2 ⟶ 1 1 \longrightarrow2\mathbb{Z} \longrightarrow\mathbb{Z} \longrightarrow\mathbb{Z}/2 \longrightarrow1 1 ⟶ 2 Z ⟶ Z ⟶ Z /2 ⟶ 1 does not split. Since 2 Z 2\mathbb{Z} 2 Z Z / 2 ∈ C F J \mathbb{Z}/2 \in\mathrm{CFJ} Z /2 ∈ CFJ Corollary 7.29 shows that ( Z , 2 Z ) (\mathbb{Z},2\mathbb{Z}) ( Z , 2 Z ) C F J \mathrm{CFJ} CFJ
The above example should be viewed as a consistency check for the normal-pair machinery rather than as a new Farrell–Jones case, since the resulting groups are also accessible by finite-cover arguments.
Example 7.46 (Surface-boundary pairs). Let Σ g , 1 \Sigma_{g,1} Σ g , 1 g g g
G = π 1 ( Σ g , 1 ) ≅ F 2 g , H = π 1 ( ∂ Σ g , 1 ) ≅ Z . G = \pi_1(\Sigma_{g,1}) \cong F_{2g}, \qquad H = \pi_1(\partial\Sigma_{g,1}) \cong\mathbb{Z}. G = π 1 ( Σ g , 1 ) ≅ F 2 g , H = π 1 ( ∂ Σ g , 1 ) ≅ Z . The subgroup H ≤ G H \leq G H ≤ G Question 7.39 . Determining whether such boundary inclusions are FJ-admissible would produce further Farrell–Jones and rigidity results for polyhedral products built from manifolds with boundary.
References 