On dimensions of groups with cocompact classifying spaces for proper actions

Ian Leary  Nansen Petrosyan

April 10, 2015

Abstract

We construct groups G that are virtually torsion-free and have virtual cohomological dimension strictly less than the minimal dimension for any model for \underline{E}G, the classifying space for proper actions of G. They are the first examples that have these properties and also admit cocompact models for \underline{E}G. We exhibit groups G whose virtual cohomological dimension and Bredon cohomological dimension are two that do not admit any 2dimensional contractible proper G-CW-complex.

1 Introduction

If G is a virtually torsion-free group, the virtual cohomological dimension vcdG, is defined to be the cohomological dimension of a torsion-free finite-index subgroup H ≤ G; a lemma due to Serre shows that this is well defined [7, VIII.3.1]. Now suppose that X is a contractible G-CW-complex that is proper, in the sense that all cell stabilizers are finite. In this case any torsion-free subgroup H will act freely on X and so X/H is a classifying space or Eilenberg-Mac Lane space BH for H. In particular, vcdG provides a lower bound for the dimension of any such X. K. S. Brown asked whether this lower bound is always attained [6, ch. 2] or [7, VIII.11]:

Does every virtually torsion-free group G admit a contractible proper G-CW-complex of dimension vcdG?

Until now, this form of Brown’s question has remained unanswered. We give examples of groups G with vcdG = 2 that do not admit any 2-dimensional contractible proper G-CW-complex in Theorem 1.3 below.

One reason why this question has been so elusive is that there are many different equivariant homotopy types of contractible proper G-CW-complexes. The most natural example is the classifying space for proper G-actions, \underline{E}G, which plays the same role in the homotopy category of proper G-CW-complexes as EG plays for free G-CW-complexes. A model for \underline{E}G is a proper G-CW complex X such that for any finite F ≤ G, the F -fixed point set X^{F} is contractible. Such an X always exists, and is unique up to equivariant homotopy equivalence. Let \underline{gd}G denote the minimal dimension of any model for \underline{E}G.

The version of Brown’s question that concerns \underline{E}G [6, ch. 2] or [7, VIII.11] is usually asked in the form: does \underline{gd}G = vcdG for every virtually torsion-free G? We prefer to split this into two separate questions. There is an algebraic dimension \underline{cd}G that bears a close relationship to \underline{gd}G, analogous to the relationship between cohomological dimension and the minimal dimension of an Eilenberg-Mac Lane space. It can be shown that \underline{cd}G = \underline{gd}G except that there may exist G for which \underline{cd}G = 2 and \underline{gd}G = 3, and \underline{cd}G is an upper bound for the cohomological dimension of any torsion-free subgroup of G [22]. In view of this we may split Brown’s question into two parts, one geometric and one algebraic.

Does there exist G for which \underline{gd}G ≠ \underline{cd}G?

Does there exist virtually torsion-free G for which \underline{cd}G > vcdG?

Examples of virtually torsion-free groups G for which \underline{gd}G = 3 and \underline{cd}G = 2 were given in [2]. These groups G are Coxeter groups. Examples of G for which \underline{cd}G > vcdG were given in [17], and more recently in [23, 12]. The advantage of the examples in [23, 12] is that in some sense they have the least possible torsion. For any virtually torsion-free G, it can be shown that \underline{cd}G is bounded by the sum vcdG + ℓ(G), where ℓ(G) is the maximal length of a chain of nontrivial finite subgroups of G [19, 6.4]. This bound is attained for the examples in [23, 12] but not for the examples in [17]. To date, all constructions of groups G for which \underline{cd}G > vcdG have used finite extensions of Bestvina-Brady groups [1], and none of these groups G admit a cocompact model for \underline{E}G. One of our main results is the construction of virtually torsion-free G admitting a cocompact \underline{E}G for which \underline{cd}G > vcdG. Amongst our examples, the easiest to describe are extensions of a right-angled Coxeter group by a cyclic group of prime order. By taking instead a cyclic extension of a torsion-free finite index subgroup of the same Coxeter group we obtain examples with cocompact \underline{E}G and for which \underline{cd}G = vcdG + ℓ(G).

Many of the terms used in the statements of our main theorems will be defined below.

Theorem 1.1. Let L be a finite n-dimensional acyclic flag complex with an admissible simplicial action of a finite group Q, and let W_{L} be the corresponding right-angled Coxeter group so that Q acts as a group of automorphisms of W_{L}. Let N be any finite-index normal subgroup of W_{L} such that N is normalized by Q, and let G be the semidirect product N ⋊ Q. This G admits a cocompact model for \underline{E}G. Let L^{sing} denote the subcomplex of L consisting of points with non-trivial stabilizer.

If H^{n}(L, L^{sing}) ≠ 0, then \underline{cd}G = n + 1 > vcdG = n.

Now suppose that L_{i}, Q_{i}, n_{i}, N_{i}, W_{i} and G_{i} are as above for i = 1, …, m, and let Γ := G_{1} × ⋯ × G_{m}. As before, there is a cocompact model for \underline{E}Γ, and

if ⨂_{i=1}^{m} H^{n_{i}}(L_{i}, L_{i}^{sing}) ≠ 0, then \underline{cd}Γ = m + ∑_{i=1}^{m} n_{i} = m + vcdΓ.

Corollary 1.2. For each m ≥ 1 there exists a virtually torsion-free group Γ_{m} admitting a cocompact \underline{E}Γ_{m} and such that

\underline{cd}Γ_{m} = 3m > vcdΓ_{m} = 2m.

For each m ≥ 1 there exists a virtually torsion-free group Λ_{m} admitting a cocompact \underline{E}Λ_{m} and such that

\underline{cd}Λ_{m} = 4m = vcdΛ_{m} + ℓ(Λ_{m}) > vcdΛ_{m} = 3m.

Furthermore, Λ_{m} may be chosen so that either every finite subgroup is cyclic or, for any fixed prime q, every nontrivial finite subgroup is abelian of exponent q.

In contrast to the above results, Degrijse and Martínez-Pérez have shown that vcdG = \underline{cd}G for a large class of groups that contains all (finitely generated) Coxeter groups [11].

Theorem 1.3. Suppose that L is a finite 2-dimensional acyclic flag complex such that the fundamental group of L admits a non-trivial unitary representation ρ : π_{1}(L) → U(n) for some n. Then vcdW_{L} = \underline{cd}W_{L} = 2, there is a cocompact 3-dimensional model for \underline{E}W_{L}, and yet there exists no proper 2-dimensional contractible W_{L}-CW-complex.

A different argument was used in [2] to show that \underline{cd}W_{L} = 2 < \underline{gd}W_{L} = 3 for some of the flag complexes L that appear in Theorem 1.3.

We remark that finitely generated Coxeter groups are linear over ℤ [8], and that this property passes to subgroups and to finite extensions. Hence all of the groups appearing in the above statements are linear over ℤ. As will be seen from the proofs, each group appearing in our statements acts properly and cocompactly on a CAT(0) cube complex; in particular they are all CAT(0) groups. A right-angled Coxeter group W_{L} is (Gromov) hyperbolic if and only if L satisfies the flag-no-square condition. Hyperbolicity passes to finite index subgroups and finite extensions. Since any 2- or 3-dimensional flag complex admits a flag-no-square subdivision [13, 24] it follows that the groups Γ_{1} and Λ_{1} in Corollary 1.2 and the groups W_{L} in Theorem 1.3 may be taken to be hyperbolic (and CAT(−1), a possibly stronger property) in addition to their other stated properties.

2 Classifying spaces and Bredon cohomology

The algebraic analogs of the geometric finiteness properties exhibited by classifying spaces of groups for families of subgroups are formulated using Bredon cohomology. This cohomology theory was introduced by Bredon in [3] for finite groups and was generalised to arbitrary groups by Lück (see [18]).

Let G be a discrete group. A family F of subgroups of G is a non-empty set of subgroups which is closed under conjugation and taking subgroups, in the sense that if H ∈ F, g ∈ G and K ≤ H, then K ∈ F and gHg^{−1} ∈ F. The orbit category O_{F}G is the category with objects the left cosets G/H for all H ∈ F and morphisms all G-equivariant functions between the objects. In O_{F}G, every morphism φ : G/H → G/K is completely determined by φ(H), since φ(xH) = xφ(H) for all x ∈ G. Moreover, there exists a morphism

G/H → G/K : H ↦ xK

if and only if x^{−1}Hx ⊆ K.

An O_{F}G-module is a contravariant functor M : O_{F}G → ℤ-mod. The category of O_{F}G-modules is denoted by Mod-O_{F}G. By definition, it has as objects all O_{F}G-modules and as morphisms all natural transformations between these objects. The category Mod-O_{F}G is an abelian category that contains enough projectives and so one can construct bi-functors Ext_{O_{F} G}^{n}(−, −) that have all the usual properties. The n-th Bredon cohomology of G with coefficients M ∈ Mod-O_{F}G is by definition

H_{F}^{n}(G; M) = Ext_{O_{F} G}^{n}(\underline{ℤ}, M),

where \underline{ℤ} is the constant functor, which sends each object to ℤ and each morphism to the identity map on ℤ. There is also a notion of Bredon cohomological dimension of G for the family F, denoted by cd_{F}(G) and defined by

cd_{F}(G) = sup{n ∈ ℕ | ∃M ∈ Mod-O_{F}G : H_{F}^{n}(G; M) ≠ 0}.

When F is the family of finite subgroups, then H_{F}^{∗}(G, M) and cd_{F}G are denoted by \underline{H}^{∗}(G, M) and \underline{cd}G, respectively. Since the augmented cellular chain complex of any model for E_{F}G yields a projective resolution of \underline{ℤ} that can be used to compute H_{F}^{∗}(G; −), it follows that cd_{F}(G) ≤ gd_{F}(G). Moreover, it is known (see for example [22, 0.1]) that

cd_{F}(G) ≤ gd_{F}(G) ≤ max{3, cd_{F}(G)}.

For any ℤG-module M, one may define an O_{F}G-module \underline{M} by \underline{M}(G/H) = Hom_{G}(ℤ[G/H], M); note that this is compatible with the notation \underline{ℤ} introduced earlier. For any G-CW-complex X with stabilizers in F, it can be shown that Bredon cohomology with coefficients in \underline{M} is naturally isomorphic to the ordinary equivariant cohomology of X with coefficients in M: H_{F}^{∗}(X; \underline{M}) ≅ H_{G}^{∗}(X; M). This follows because there is an isomorphism of cochain complexes

Hom_{F}(C_{∗}^{F}(X), \underline{M}) ≅ Hom_{G}(C_{∗}(X), \underline{M}).

A subfamily of a family F of subgroups of G is another family G ⊆ F. For a subfamily G and a G-CW-complex X with stabilizers in F, the G-singular set X^{G-sing} is the subcomplex consisting of points of X whose stabilizer is not contained in G. When G consists of just the trivial subgroup this is the usual singular set and we write X^{sing} for X^{G-sing}.

Given a ℤG-module M and a subfamily G of F, we define two further O_{F}Gmodules: a submodule \underline{M}_{>G} of \underline{M}, and the corresponding quotient module \underline{M}_{≤G}. These are defined by

\underline{M}_{>G} : G/H ↦ \{ \begin{matrix}Hom_{G}(ℤ[G/H], M) & if & H ∉ G, \\ 0 & if & H ∈ G, \end{matrix}

\underline{M}_{≤G} : G/H ↦ \{ \begin{matrix}0 & if & H ∉ G, \\ Hom_{G}(ℤ[G/H], M) & if & H ∈ G. \end{matrix}

By construction there is a short exact sequence of O_{F}G-modules

\underline{M}_{>G} ↣ \underline{M} ↠ \underline{M}_{≤G}.

Hence there is a short exact sequence of cochain complexes.

0 → Hom_{F}(C_{∗}^{F}(X), \underline{M}_{>G}) → Hom_{F}(C_{∗}^{F}(X), \underline{M}) → Hom_{F}(C_{∗}^{F}(X), \underline{M}_{≤G}) → 0

This gives rise to a long exact sequence in Bredon cohomology. Since there is a natural identification between Bredon cohomology with coefficients in \underline{M}_{>G} and the equivariant cohomology of X^{G-sing} with coefficients in M:

H_{F}^{∗}(X; \underline{M}_{>G}) ≅ H_{G}^{∗}(X^{G-sing}; M),

we deduce that the Bredon cohomology with coefficients in \underline{M}_{≤G} is isomorphic to the equivariant cohomology of the pair (X, X^{G-sing}) with coefficients in M:

H_{F}^{∗}(X; \underline{M}_{≤G}) ≅ H_{G}^{∗}(X, X^{G-sing}; M).

Hence we obtain the following.

Proposition 2.1. Let F be a family of subgroups of G, with G a subfamily, let X be any model for E_{F}G, and let H be a finite-index subgroup of G. There exists a Bredon module C such that the Bredon cohomology of the group G with coefficients in C computes the ordinary cohomology of the pair (X/H, X^{G-sing}/H):

H_{F}^{∗}(G; C) ≅ H^{n}(X/H, X^{G-sing}/H; ℤ).

Furthermore, each abelian group C(G/K) is finitely generated.

Proof. Let M be the permutation module ℤ[G/H], and let C := \underline{M}_{≤G}. For each K ∈ F, C(G/K) is isomorphic to a subgroup of ℤ[G/H] and so is finitely generated, and

H_{F}^{∗}(G; C) ≅ H_{F}^{∗}(X; C) ≅ H_{G}^{∗}(X, X^{G-sing}; ℤ[G/H])
≅ H_{H}^{∗}(X, X^{G-sing}; ℤ) ≅ H^{∗}(X/H, X^{G-sing}/H; ℤ),

where the first two isomorphisms follow from the discussion above and the third because H has finite index in G.

3 Right-angled Coxeter groups

A right-angled Coxeter system consists of a group W called a right-angled Coxeter group together with a set S of involutions that generate W, subject to only the relations that certain pairs of the generators commute. We will always assume that S is finite. The right angled Coxeter system is determined by the graph L^{1}(W, S) with vertex set S and edges those pairs of vertices that commute. Equivalently, the right-angled Coxeter system is determined by the flag complex L(W, S) with vertex set S and simplices the cliques in the graph L^{1}(W, S).

Given a right-angled Coxeter system (W, S), the Davis complex Σ(W, S) can be realized as either a cubical complex, or as a simplicial complex which is the barycentric subdivision of the cubical complex. For more details concerning the Davis complex, we refer the reader to [10] or [9]. The simplicial structure is easier to describe, so we consider this first. A spherical subset T of S is a subset whose members all commute; equivalently T is either the empty set, or a subset of S that spans a simplex of L(W, S). A special parabolic subgroup of W is the subgroup of W generated by a spherical subset T. We denote the special parabolic subgroup generated by T by W_{T}. A parabolic subgroup of W is a conjugate of a special parabolic subgroup. The set of cosets of all special parabolic subgroups forms a poset, ordered by inclusion, and the simplicial complex Σ(W, S) is the realization of this poset. By construction, W acts admissibly simplicially on Σ(W, S) in such a way that each stabilizer subgroup is parabolic.

If T is a spherical subset of S, then the subposet of cosets contained in W_{T} is equivariantly isomorphic to the poset of faces of the standard |T |-cube [−1, 1]^{T}, with the group W_{T} ≅ C_{2}^{T} acting via reflections in the coordinate hyperplanes. In this way we obtain a cubical structure on Σ(W, S), in which the n-dimensional subcubes correspond to cosets wW_{T} with |T | = n. The setwise stabilizer of the cube wW_{T} is the parabolic subgroup wW_{T} w^{−1}, which acts on the cube in such a way that the natural generators wtw^{−1} act as reflections in the coordinate hyperplanes. The simplicial complex described above is the barycentric subdivision of this cubical complex.

If we view every simplicial complex as containing a unique −1-simplex corresponding to the empty subset of its vertex set, then we get a natural bijective correspondence between the W -orbits of cubes in Σ(W, S) and the simplices of L(W, S) which preserves incidence (the empty simplex corresponds to the 0-cubes). Hence we obtain:

Proposition 3.1. There is a natural bijection between subcomplexes of the simplicial complex L(W, S) and non-empty W -invariant subcomplexes of the cubical complex Σ(W, S).

To show that Σ(W, S) is a model for \underline{E}W, metric techniques are helpful. There is a natural CAT(0)-metric on Σ(W, S), which is best understood in terms of the cubical structure. The length of a piecewise linear path in Σ(W, S) is defined using the standard Euclidean metric on each cube, and the distance between two points of Σ(W, S) is the infimum of the lengths of PL-paths connecting them. According to Gromov’s criterion [15], Σ(W, S) is locally CAT(0) because the link of every vertex is isomorphic to L(W, S) which is a flag complex (see [10, 15]). It is easy to see that Σ(W, S) is simply connected (for example, because its 2-skeleton is a version of the Cayley complex for W), and it follows that Σ(W, S) is CAT(0) [5, Theorem II.4.1]. Given that W acts isometrically with finite stabilizers on Σ(W, S) it follows that Σ(W, S) is a model for \underline{E}W via the Bruhat-Tits fixed point theorem [5, p. 179] or [2, Prop. 3].

Lemma 3.2. Every finite subgroup of W is a subgroup of a parabolic subgroup of W. In particular, there are finitely many conjugacy classes of finite subgroups of W and every finite subgroup is isomorphic to a direct product (ℤ/2)^{k} for some 0 ≤ k ≤ n where n is the dimension of Σ(W, S).

Proof. Let F be a finite subgroup of W. By the Bruhat-Tits fixed point theorem F fixes some point of Σ, and hence F is a subgroup of a point stabilizer. Every such subgroup is parabolic, and each is conjugate to one of the finitely many special parabolics.

Corollary 3.3. The commutator subgroup W′ of W is a finite-index torsionfree subgroup of type F.

Proof. The abelianization of W is naturally isomorphic to C_{2}^{S}. Every parabolic subgroup of W maps injectively into C_{2}^{S}. It follows that W′ acts freely on the finite-dimensional contractible space Σ. Hence Σ/W′ is a compact K(W′, 1), from which it follows that W′ is both type F and torsion-free.

Lemma 3.4 ([2]). The quotient of the pair (Σ, Σ^{sing}) by W is isomorphic to the pair (CL′, L′), i.e., the pair consisting of the cone on the barycentric subdivision of L and its base. This isomorphism is natural for automorphisms of L. If L is acyclic then so is Σ^{sing}. If L is simply-connected, then so is Σ^{sing}.

Proof. The first part is clear from the simplicial description of Σ. Now let V be the unique free W -orbit of vertices in the simplicial description of Σ. The star of each v ∈ V is a copy of the cone CL′, with v as its apex. The subcomplex of Σ consisting of all simplices not containing any vertex of V is Σ^{sing}. Hence Σ is obtained from Σ^{sing} by attaching cones to countably many subcomplexes isomorphic to L′.

In the case when L is acyclic, attaching a cone to a copy of L′ does not change homology. It follows that Σ^{sing} must be acyclic since Σ is. Similarly, if L is simply-connected, then attaching a cone to a copy of L′ does not change the fundamental group, so Σ^{sing} must be simply-connected since Σ is.

Now suppose a finite group Q acts by automorphisms on L(W, S). This defines an action of Q on W, and hence a semidirect product G = W ⋊ Q.

Lemma 3.5. There is an admissible simplicial G-action on Σ(W, S) extending the action of W, and Σ(W, S) becomes a cocompact model for \underline{E}G.

Proof. The action of Q on the poset underlying Σ(W, S) is defined in such a way that q ∈ Q sends the coset wW_{T} to the coset q(w)W_{q(T)}. This combines with the W -action to give an admissible G-action on Σ(W, S). Since Σ(W, S) is CAT(0) and the stabilizers are finite it follows that Σ(W, S) is a model for \underline{E}G.

Corollary 3.6. Any finite-index subgroup H of G as above admits a cocompact model for \underline{E}H and is virtually torsion-free.

Remark 3.7. For the action of G on Σ, the stabilizer of the vertex W_{T} is the semidirect product W_{T} ⋊ Q_{T}, where Q_{T} := {q ∈ Q : q(T) = T }. If Q acts admissibly on L then Q_{T} fixes each element of T and the stabilizer is the direct product W_{T} ×Q_{T}. Similarly, the stabilizer of the vertex wW_{T} is the direct product wW_{T} w^{−1} ×wQ_{T} w^{−1}. Note in particular that the image of the stabilizer under the quotient map G → G/W ≅ Q depends only on T, and not on w.

Let W^{ev} denote the subgroup of elements of W expressible as words of even length in the elements of S. Thus W^{ev} is an index two subgroup of W.

Lemma 3.8. Let N be a finite index normal subgroup of W^{ev}. There is an isomorphism ψ from the relative chain complex C_{∗}(CL′, L′) to a direct summand of the simplicial chain complex C_{∗}(Σ/N). This isomorphism is natural for automorphisms of L that preserve N. It is also natural for the inclusion of subcomplexes in L and W/N-invariant subcomplexes of the cubical structure on Σ/N.

Proof. The cone CL′ is the realization of the poset of spherical subsets of S, with cone point the empty set ∅. For σ a simplex of CL′, ψ(σ) in C_{∗}(Σ/N) will be the signed sum of its |W/N| inverse images under the map Σ/N → Σ/W = CL′. The signs will ensure that simplices of CL′ that do not contain ∅ as a vertex map to zero.

In more detail, fix a transversal w_{1}, …, w_{m} to N in W, and for σ a simplex of CL′, viewed as a chain σ = (T_{0} < T_{1} < ⋯ < T_{r}) of spherical subsets, define

ψ(σ) = ∑_{i=1}^{m}(−1)^{l(w_{i})}w_{i}σ = ∑_{i=1}^{m}(−1)^{l(w_{i})}(w_{i}W_{T_{0}} < ⋯ < w_{i}W_{T_{r}}).

Here l(w) denotes the length of w as a word in S. For any n ∈ N, l(wn) − l(w) is even, and so the sum above does not depend on the choice of transversal. The above formula clearly describes a chain map from C_{∗}(CL′) to C_{∗}(Σ/N). Now if T is a non-empty spherical subset of S, W_{T} contains equal numbers of words of odd and even length, and hence so does its image W_{T} /(W_{T} ∩ N) ≤ W/N. It follows that if T_{0} ≠ ∅, then ψ(σ) = 0. Hence the formula given above defines a chain map ψ : C_{∗}(CL′, L′) → C_{∗}(Σ/N). This clearly has the claimed naturality properties.

It remains to exhibit a splitting map ϕ : C_{∗}(Σ/N) → C_{∗}(CL′, L′). This uses a ‘simplicial excision map’. Let v be the image of W_{∅} ∈ Σ in Σ/N, and let X be the subcomplex of Σ/N consisting of all simplices that do not have v as a vertex. There is a natural bijection between simplices of CL′ containing the cone vertex and simplices of Σ/N containing v. This induces an isomorphism C_{∗}(Σ/N, X) ≅ C_{∗}(CL′, L′) and ϕ is defined as the composite of this with the map C_{∗}(Σ/N) → C_{∗}(Σ/N, X).

Corollary 3.9. With notation as above, let K be a subcomplex of L, and let Σ(K) be the (barycentric subdivision of the) cubical W_{L}-subcomplex of Σ associated to K. There is a natural isomorphism ψ from the relative chain complex C_{∗}(CL′, L′ ∪ CK′) to a direct summand of the relative simplicial chain complex C_{∗}(Σ/N, Σ(K)/N).

Proof. C_{∗}(CK′, K′) is a subcomplex of C_{∗}(CL′, L′) and the corresponding quotient is C_{∗}(CL′, L′ ∪CK′). Similarly, C_{∗}(Σ(K)/N) is a subcomplex of C_{∗}(Σ/N) with C_{∗}(Σ/N, Σ(K)/N) the corresponding quotient.

By naturality of ψ and ϕ we get a diagram as follows, in which the two left-hand squares with the same label on both vertical sides commute and such that the two composites labelled ϕ ∘ ψ are equal to the relevant identity maps. A diagram chase shows that there are unique maps ψ and ϕ corresponding to the dotted vertical arrows that make the right-hand squares with the same label on both vertical sides commute, and that these maps also satisfy ϕ ∘ ψ = 1.

\xymatrix{0 \ar[r] & C_{∗}(CK, K′) \ar[d]^{ψ} \ar[r] & C_{∗}(CL′, L′) \ar[d]^{ψ} \ar[r] & C_{∗}(CL′, L′ ∪ CK′) \ar@{-->}[d]^{ψ} \ar[r] & 0 \\ 0 \ar[r] & C_{∗}(Σ(K)/N) \ar[u]^{ϕ} \ar[r] & C_{∗}(Σ/N) \ar[u]^{ϕ} \ar[r] & C_{∗}(Σ/N, Σ(K)/N) \ar@{-->}[u]^{ϕ} \ar[r] & 0}

4 Proof of Theorem 1.1

As in the statement of Theorem 1.1, let L be a finite n-dimensional flag complex equipped with an admissible simplicial action of a finite group Q, let (W, S) = (W_{L}, S_{L}) be the associated right-angled Coxeter system, let N be a finite-index subgroup of W that is normalized by Q, and let G be the semidirect product G = N ⋊ Q.

Proof of Theorem 1.1. Note that G may be viewed as a finite-index subgroup of the semidirect product W ⋊ Q. Under these hypotheses, we already see from Corollary 3.5 that the Davis complex Σ is a cocompact (n + 1)-dimensional model for \underline{E}G, and that G is virtually torsion-free.

Using the hypothesis that L is acyclic, we see that the subcomplex Σ^{sing(W)} of Σ consisting of those points whose stabilizer in W is non-trivial is acyclic by Lemma 3.4. In this case any finite-index torsion-free subgroup of G acts freely on the acyclic n-dimensional complex Σ^{sing(W)}, which implies that vcdG ≤ n. The fact that equality holds can be deduced from the calculation of the cohomology of W with free coefficients in Theorem [10, 8.5.2], but we also give an alternative proof. Let K := L^{sing}, the Q-singular points in L. From the long exact sequence for the pair (L, K) we see that H^{n−1}(K) ≠ 0, and hence H^{n}(CK, K) ≠ 0. Lemma 3.8 applied to the Coxeter group W_{K} and its finite-index torsion-free subgroup W′_{K} shows that H^{n}(W′_{K}; ℤ) = H^{n}(Σ_{K}/W′_{K}) contains a summand isomorphic to H^{n}(CK, K) and so is not zero. (Here we use Σ_{K} to denote Σ(W_{K}, S_{K}) since we reserve Σ to stand for Σ(W_{L}, S_{L}).) Hence vcdG ≥ vcdW_{K} ≥ n.

It remains to show that \underline{cd}G ≥ n + 1. Since W^{ev} has index 2 in W and is clearly Q-invariant, we see that (N ∩ W^{ev}) ⋊ Q is a subgroup of G of index at most 2. Hence without loss of generality we may assume that N ≤ W^{ev}. Now consider the family W of finite subgroups of G, consisting of those finite subgroups that are contained in N, or equivalently the finite subgroups that map to the trivial subgroup under the factor map G → Q. The stabilizers in W ⋊ Q of vertices of Σ are described in Remark 3.7, and by intersecting with G = N ⋊Q we get a similar description of stabilizers in G: the stabilizer of the vertex wW_{T} is the direct product of the intersection N ∩wW_{T} w^{−1} and a subgroup that maps isomorphically to Q_{T}, the stabilizer in Q of the vertex T of CL′. It follows that Σ^{W-sing} is equal to the inverse image in Σ of the Q-singular set CK′ in Σ/W = CL′. Hence Σ^{W-sing} is the W -invariant subcomplex of the cubical structure on Σ that corresponds (under the map of Proposition 3.1) to K := L^{sing}. Using Corollary 3.9 applied in this case, we see that H^{n+1}(Σ/N, Σ^{W-sing}/N) admits a split surjection onto H^{n+1}(CL′, L′ ∪ CK′), which is isomorphic to H^{n}(L, K) = H^{n}(L, L^{sing}) since L is acyclic. Proposition 2.1 finishes the argument.

The general case of Theorem 1.1 follows from the case described above. Let each G_{i} be defined as above in terms of L_{i}, Q_{i}, n_{i} and N_{i} ≤ W_{i}, and define Γ := G_{1} × ⋯ × G_{m}, Q = Q_{1} × ⋯ × Q_{m}, and W := W_{1} × ⋯ × W_{m}. Finally, let n := ∑_{i=1}^{m} n_{i}. The direct product Σ := Σ_{1} × ⋯ × Σ_{m} is a cocompact model for \underline{E}Γ of dimension m + n, and so \underline{cd}Γ ≤ m + n. Also the direct product Σ_{1}^{sing} × ⋯ × Σ_{m}^{sing} is an acyclic n-dimensional simplicial complex admitting a proper Γ-action, which implies that vcdΓ ≤ n. The lower bounds also work just as in the case m = 1. To give a lower bound for vcdΓ, let K_{i} := L_{i}^{sing}, and note that H^{n_{i}−1}(K_{i}) ≅ H^{n_{i}}(L_{i}, K_{i}). The group Γ contains a subgroup Γ′ that is isomorphic to a finite-index subgroup of the product W′ := W′_{1} × ⋯ × W′_{m} of the commutator subgroups. By the universal coefficient theorem H^{n}(W′; ℤ) is non-zero, since it contains a summand isomorphic to ⨂_{i=1}^{m} H^{n_{i}}(CK_{i}, K_{i}) ≠ {0}. If we define W to be the family of subgroups of Γ that are contained in W, then a point x = (x_{1}, …, x_{m}) ∈ Σ = Σ_{1} × ⋯ × Σ_{m} is in Σ^{W-sing} if and only if there is an i so that x_{i} ∈ Σ_{i}^{W_{i}-sing}. Hence we see that if we define N := N_{1} ×⋯×N_{m}, then

H^{n+m}(Σ/N, Σ^{W-sing}/N) ≅ ⨂_{i=1}^{m} H^{n_{i}+1}(Σ_{i}/N_{i}, Σ_{i}^{W_{i}-sing}/N_{i}),

the latter of which is non-zero since it contains a summand isomorphic to

⨂_{i=1}^{m} H^{n_{i}+1}(CL_{i}, L_{i} ∪ CK_{i}) ≅ ⨂_{i=1}^{m} H^{n_{i}}(L_{i}, K_{i}) = ⨂_{i=1}^{m} H^{n_{i}}(L_{i}, L_{i}^{sing}).

5 Applications

In this section we describe various pairs (Q, L) consisting of a finite group Q and an admissible Q-action on a flag complex L, producing sufficiently many examples to establish Corollary 1.2. Since any Q-CW-complex is equivariantly homotopy equivalent to a flag simplicial complex with an admissible action of Q, we do not give explicit triangulations of L. Example 1. Let Q be the alternating group A_{5}, and define a Q-CW-complex as follows. For the 1-skeleton L^{1} of L take the complete graph on five vertices, with the natural action of Q = A_{5}. In A_{5}, the 24 elements of order five split into two conjugacy classes of size 12, and any element g of order 5 is conjugate to g^{−1} (but is not conjugate to g^{2} or g^{3}). Define L by using one of the two conjugacy classes of 5-cycles to describe attaching maps for six pentagonal 2cells. By construction there is a Q-action without a global fixed point and it is easily checked that L is acyclic. In fact, π_{1}(L) is isomorphic to SL(2, 5), the unique perfect group of order 120, and L is isomorphic to the 2-skeleton of the Poincaré homology sphere [3, I.8]. By taking each Q_{i} to be A_{5} and each L_{i} to be a flag triangulation of L as defined above, we obtain groups Γ_{m} having the properties stated in the first part of 1.2. Example 2. Fix distinct primes p and q, and let Q be cyclic of order q, generated by g. For the Q-fixed point set L^{Q}, take a mod-p Moore space M(1, p). This space has a CW-structure with 1-skeleton a circle and one 2-cell f. The 2-cell f is attached to the circle via a map of degree p. Now define L^{2} by adding on a free Q-orbit of 2-cells f_{0}, …, f_{q−1}, where f_{i} = g^{i}f_{0}, so that each f_{i} is attached to the circle by a degree one map. L^{2} is simply connected, and H_{2}(L^{2}) is a free ℤQ-module of rank one, since it has a ℤ-basis given by the elements

e_{j} := f − ∑_{i=0}^{p} g^{i}f_{j} = f − ∑_{i=0}^{p} g^{i+j}f_{0}

for 0 ≤ j < q, and g^{j}e_{0} = e_{j} for each j. Make L by attaching a free Qorbit of 3-cells to kill each e_{j}. To establish the second part of Corollary 1.2 we take each Q_{i} to be cyclic of order q_{i}, take each L_{i} to be as above for some fixed choice of p, and take N_{i} to be the commutator subgroup of the Coxeter group W_{i} := W (L_{i}, S_{i}). For any such choice, we obtain a group Λ_{m} as in the statement. To ensure that Λ_{m} contains only cyclic finite subgroups we must take the primes q_{i} all distinct, whereas to ensure that Λ_{m} contains only abelian finite subgroups of exponent q we take q_{i} = q for all i.

6 Contractibility and acyclicity

In [2], it was shown that certain right-angled Coxeter groups W have the property that vcdW = \underline{cd}W = 2 < \underline{gd}W = 3. In this section we improve this result by showing that for these same groups there is no 2-dimensional contractible proper W -CW-complex.

We will use a few subsidiary results in the proof.

Proposition 6.1. If Y is a subcomplex of a 2-dimensional acyclic complex, then H_{2}(Y) = 0 and each H_{i}(Y) is free abelian.

Proof. If Y is any subcomplex of an n-dimensional acyclic complex Z, then consideration of the homology long exact sequence for the pair (Z, Y) shows that H_{n}(Y) is trivial and that H_{n−1}(Y) is free abelian. Since H_{0} is always free abelian, the case n = 2 gives the claimed result.

Proposition 6.2. Let Q be a finite soluble group and let X be a 2-dimensional acyclic Q-CW-complex. Then the fixed point set X^{Q} is also acyclic.

Proof. The finite soluble group Q has a normal subgroup N of prime index, the factor group Q/N acts on the N-fixed point set X^{N}, and the equality X^{Q} = (X^{N})^{Q/N} holds. Hence it suffices to consider the case in which Q has prime order.

By the P. A. Smith theorem, X^{Q} is mod-p acyclic in the case when Q has order p. By the previous proposition, H_{i}(X^{Q}) is free abelian for all i. By the universal coefficient theorem, the rank of the ith mod-p homology group of X^{Q} is equal to the rank of H_{i}(X^{Q}). Hence X^{Q} must be acyclic.

Lemma 6.3. Let X be a CW-complex, let S be a finite indexing set, and let X(s) be a subcomplex of X such that each X(s) is acyclic and each intersection of X(s)’s is either empty or acyclic. Define

X^{#} := ⋃_{s∈S} X(s),

and let |N | be the realization of the nerve of the covering of X^{#} by the subcomplexes X(s). There is a map f : X^{#} → |N | which is a homology isomorphism and induces a surjection of fundamental groups.

Proof. In the case when each intersection of X(s)’s is either contractible or empty, it is well-known that there is a homotopy equivalence f : X^{#} → |N | [16, 4.G, Ex. 4]. We use Quillen’s plus construction to reduce to this case.

For T ⊆ S, define X(T) to be the intersection X(T) = ⋂_{s∈T} X(s). Suppose that U ⊆ S is such that X(U) is non-empty. In this case, since X(U) is acyclic we can find a set A_{U} of 2-cells with attaching maps from the boundary of the 2-cell to X(U) so that each attaching map represents a conjugacy class of commutators in π_{1}(X(U)) and so that the fundamental group of the resulting complex X̂(U) is trivial. Moreover, there is a set B_{U} of 3-cells and attaching maps from the boundary of the 3-cell to X̂(U) so that the resulting complex X_{U} contains X(U) as a subcomplex, is simply-connected, and such that the inclusion of X(U) into X_{U} is a homology isomorphism. Define Y by attaching to X 2- and 3-cells indexed by ∐_{U} A_{U} and ∐_{U} B_{U} respectively. Define a subcomplex Y (s) of Y by attaching to X(s) the 2- and 3-cells indexed by ∐_{s∈U} A_{U} and ∐_{s∈U} B_{U} respectively. Finally define Y (T) := ⋂_{s∈T} Y (s), and Y^{#} := ⋃_{s∈S} Y (s). The nerve of the covering of Y^{#} by the subcomplexes Y (s) is naturally isomorphic to N. A Mayer-Vietoris spectral sequence argument shows that the inclusion X^{#} → Y^{#} is a homology isomorphism, and this map induces a surjection π_{1}(X^{#}) → π_{1}(Y^{#}) because the 1-skeleta of X^{#} and Y^{#} are equal.

Proof of Theorem 1.3. The Davis complex Σ = Σ(W_{L}, S_{L}) is a cocompact 3dimensional model for \underline{E}W_{L}. Since L is acyclic, Σ^{sing} is a 2-dimensional acyclic proper W_{L}-CW-complex in which the fixed point set for any non-trivial finite subgroup is contractible. This suffices to show that \underline{cd}W_{L} = 2.

Now suppose that X is any contractible proper 2-dimensional W_{L}-CW-complex. Let S = S_{L}, and define X^{#} as the union of the fixed point sets X^{s}: X^{#} := ⋃_{s∈S} X^{s}. By construction, the nerve of the covering of X^{#} by the sets X^{s} is equal to L. By Propostion 6.2, for each T ⊆ S that spans a simplex of L the subset

X(T) := ⋂_{s∈T} X^{s} = X^{⟨T ⟩}

is acyclic, and for each T that does not span a simplex of L, X(T) is empty. By Lemma 6.3, it follows that there is a natural surjection ϕ : π_{1}(X^{#}) → π_{1}(L). Define ρ′ := ρ ∘ ϕ : π_{1}(X^{#}) → U(n), a non-trivial unitary representation of π_{1}(X^{#}). We use this representation to obtain a contradiction.

Pick g ∈ π_{1}(X^{#}) so that ρ′(g) ≠ 1. Since X is contractible, there exists connected X_{1} with X^{#} ⊆ X_{1} ⊆ X such that X_{1} − X^{#} comprises only finitely many cells, and such that g maps to the identity element of π_{1}(X_{1}). By Proposition 6.1, H_{1}(X_{1}) is free abelian. By contracting some of the 1-cells in X_{1} −X^{#} we may replace X_{1} by a complex X_{2}, with the following properties: X^{#} ⊆ X_{2}; H_{1}(X_{2}) is free abelian and H_{2}(X_{2}) = {0}; X_{2} consists of X^{#} with finitely many 1- and 2-cells added; g is in the kernel of the map π_{1}(X^{#}) → π_{1}(X_{2}). Now make X_{3} by attaching 2-cells to exactly kill H_{1}(X_{2}). Thus X_{3} is an acyclic 2-complex, obtained by attaching finitely many 1- and 2-cells to X^{#}. By the Gerstenhaber-Rothaus theorem [14], the representation ρ′ : π_{1}(X^{#}) → U(n) extends to a representation ρ̃ : π_{1}(X_{3}) → U(n). However, this contradicts the fact that ρ′(g) ≠ 1, while g maps to the identity in π_{1}(X_{3}).

Remark 6.4. As an example of a suitable L, take a flag triangulation of the 2-skeleton of the Poincaré homology sphere (which was discussed in the previous section); here there is a faithful representation ρ : π_{1}(L) ≅ SL(2, 5) → U(2).

Remark 6.5. There is a version of Brown’s question that remains open: for m > 2, is there a virtually torsion-free group G such that vcdG = m but there exists no contractible m-dimensional proper G-CW-complex?

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