SOME EXAMPLES OF DISCRETE GROUP ACTIONS ON ASPHERICAL MANIFOLDS

M. W. DAVIS AND I. J. LEARY

Abstract. We construct two classes of examples of a virtually torsion-free group G acting properly and cocompactly on a contractible manifold X. In the first class of examples the universal space for proper actions, \underline{E}G, has no model with finitely many orbits of cells. The reason is that the centralizers of certain finite subgroups of G will not have finite-type classifying spaces. In the second class of examples X is a CAT (0) manifold upon which G acts by isometries. It follows that X is a model for \underline{E}G. In these examples the fixed point sets of certain finite subgroups of G are not manifolds and the centralizers of these subgroups are not virtual Poincaré duality groups.

1. Introduction

A discrete group G is type F if its classifying space BG (also called its “Eilenberg-Mac Lane space”) has the homotopy type of a finite complex. (There is a weaker, algebraic version of this condition known as “type FP”; the definition can be found in [5].) A group of type F (or FP) is necessarily torsion-free. A virtually torsion-free group G is type VF (resp. type VFP) if it contains a finite index subgroup of type F (resp. of type FP).

For any discrete group G, there is a CW-complex \underline{E}G on which G acts cellularly and properly (i.e., with finite isotropy subgroups) so that for each subgroup H of G, the fixed set (\underline{E}G)^{H} is contractible whenever H is finite (and is empty when H is infinite). \underline{E}G is called the universal space for proper G-actions. It is unique up to G-homotopy equivalence. (See [17].) We say that G is type VF if there is an \underline{E}G so that the quotient space \underline{E}G/G is compact. A group G of type VF is

Date: February 22, 2002. 2000 Mathematics Subject Classification. Primary: 57S30; Secondary: 20F36,

20F55, 57M07.

Key words and phrases. finiteness conditions, Poincaré duality group. The first author was partially supported by NSF grant DMS-0104026. The

second author acknowledges support from EPSRC (via grant GR/R07813), and from the Ohio State Mathematical Research Institute.

not necessarily virtually torsion-free (see, for example, p. 493 of [2]), but any virtually torsion-free group of type VF is of type VF. There is also a similar, but weaker, algebraic notion of a group being of “type VF P ”, which will be defined in an appendix to this article.

In many of the most well-known examples of groups of type VF or of type VF, \underline{E}G is a contractible manifold. For example, suppose that X is a simply connected, complete Riemannian manifold of nonpositive sectional curvature (e.g., a symmetric space) and that G is a discrete group of isometries of X. For each finite subgroup H, the nonpositive curvature condition forces the fixed set X^{H} to be nonempty and convex and hence, to be contractible. The same argument works even when the metric is singular, i.e., when X is a “CAT (0) space”. (See [4] for the definitions and basic facts concerning CAT (0) spaces.) Thus, any CAT (0) X is a model for \underline{E}G. In particular, if the quotient space X/G is compact, then G is type VF. We note that, in the Riemannian case, since the G-action is smooth, each fixed set X^{H} is a smooth submanifold of X.

In [12] the second author and B. Nucinkis produced the first examples of groups of type VF which were not of type VF. How does one tell that a given group G is not of type VF ? For any finite subgroup H of G, let V_{H} = N_{G}(H)/H, where N_{G}(H) stands the normalizer of H in G. Then (\underline{E}G)^{H} is the universal space for proper V_{H}-actions and (\underline{E}G)^{H}/V_{H} is emedded as a closed subspace of \underline{E}G/G. It follows from this, first of all, that if G is type VF, then (since \underline{E}G/G is compact) there are only finitely many conjugacy classes of finite subgroups in G. Secondly, if G is type VF, then so is V_{H}. In particular, if V_{H} contains a torsion-free subgroup of finite index which is not type F, then G cannot be type VF. In [12] various examples of G of type VF are constructed which violate both these conditions. The method of [12] is to combine the construction of Bestvina-Brady [1] with earlier examples (e.g., of Oliver [16]) of exotic actions of finite groups on contractible finite complexes.

In [8] the first author showed how the reflection group trick could be used to promote the Bestvina-Brady examples to examples of Poincaré duality groups which cannot be finitely presented. In our first class of examples we show how the same trick can be used to promote the examples of [12] to examples where a virtually torsion-free group G acts cocompactly on a contractible manifold, thereby proving the following theorem.

Theorem 1. There are examples of virtually torsion-free groups G such that G acts properly and cocompactly on a contractible manifold and such that G is not of type VF.

A discrete group G is a P D^{n}-group (or a “Poincaŕe duality group”) if it is type FP and if

H^{i}(G; ℤG) = \begin{cases}ℤ, & if i = n, \\ 0, & otherwise. \end{cases}

It follows that a group of type F is a P D^{n}-group if and only if its classifying space is a Poincaré complex. A virtually torsion-free group G is a V P D^{n}-group (a “virtual Poincaré duality group”) if it contains a subgroup of finite index which is a P D^{n}-group.

Suppose that G admits a \underline{E}G which is a manifold with cocompact G-action (this of course implies that G is of type VF). If for each finite subgroup H, (\underline{E}G)^{H} is also a contractible manifold, then the cohomology with compact supports of (\underline{E}G)^{H} is concentrated in the top dimension (and is isomorphic to ℤ in that dimension). This dimension is, in fact, cd_{ℚ}(V_{H}) (where cd_{ℚ}(−) stands for “cohomological dimension over ℚ”). We also note that the cohomology with compact supports of (\underline{E}G)^{H} is isomorphic to H^{∗}(V_{H}; ℤV_{H}) (see Exercise 4, p. 209 in [5]). This motivates the following definition.

Definition. A group G is a VPD-group if it is type VF P and if for each finite subgroup H

H^{i}(V_{H}; ℤV_{H}) = \begin{cases}ℤ, & if i = cd_{ℚ}(V_{H}), \\ 0, & otherwise. \end{cases}

The above definition relies on the definition of type VF P, which we include in an appendix, as it is rather technical. We shall apply the above definition only in the case when G is type VF, which implies that G is type VF P. Hence the reader who is not interested in algebra will be able to read the rest of this article without reading the appendix.

In our second class of examples G will be a virtually torsion-free group acting cocompactly on a CAT (0) manifold X (so G is type VF ) yet G is not a VPD-group. The point is that for certain finite subgroups H, the fixed set X^{H} will not be a submanifold and will not satisfy Poincaré duality. In these examples the group G will be the semidirect product of a right-angled Coxeter group and a finite group. So, we will have proved the following.

Theorem 2. There are examples of virtually torsion-free groups G such that G is a discrete, cocompact group of isometries of a CAT(0) manifold and such that G is not a VPD-group.

2. The equivariant reflection group trick

A simplicial complex L is a flag complex if any finite set of vertices which are pairwise connected by edges spans a simplex of L.

Suppose we are given as data:

• a space M and a subspace N such that N is triangulated as a finite dimensional flag complex,

• the action of a discrete group G on M so that G stabilizes N and so that G acts on N by simplicial automorphisms.

To this data we will associate a right-angled Coxeter group W, a space U(M, N, G) and an action of the semidirect product W ⋊ G on U(M, N, G). (In practice, in this paper, M will always be a manifold with boundary and N = ∂M.)

Let I denote the vertex set of the triangulation of N. W is defined by a presentation as follows. There is one generator s_{i} for each i ∈ I. The relations are given by s_{i}^{2} = 1, for i ∈ I, and (s_{i}s_{j})^{2} = 1 whenever {i, j} spans an edge of N. The G-action on I induces an action of G on W by automorphisms. Hence, we can form the semidirect product W ⋊ G.

For each vertex i, let N_{i} denote the closed star of i in the barycentric subdivision of N. For each x ∈ N, let σ(x) = {i ∈ I|x ∈ N_{i}} and let W_{x} denote the subgroup generated by {s_{i}|i ∈ σ(x)}. The space U(M, N, G) is defined by

U(M, N, G) = (W × M)/ ∼

where ∼ is the equivalence relation defined by (w, x) ∼ (w′, x′) if and only if x = x′ and w^{−1}w′ ∈ W_{x}. The image of (w, x) in U(M, N, G) is denoted [w, x]. For [w, x] in U(M, N, G) and (v, g) in W ⋊ G, the action of W ⋊ G on U(M, N, G) is defined by

(v, g) ▪ [w, x] = [vw, gx].

Some important features of this construction are:

• If M is contractible, then so is U(M, N, G).

• If M is an n-manifold with boundary and ∂M = N, then U(M, N, G) is an n-manifold.

• If N is a finite complex and M = Cone(N), the cone on N, then U(Cone(N), N, G) has a natural CAT (0) cubical structure so that the link of each vertex is isomorphic to N and so that W ⋊ G is a group of isometries.

The proofs of these facts, in the case where G is the trivial group, can be found in [7] and [10]. (See also [8], [9] and [11].)

Let us say that the G-action on N is admissible if for every simplex σ of N, the setwise and pointwise stabilizers of σ are equal. (This can always be achieved by passing to the barycentric subdivision of N, cf., [3, p. 116].) Assume that the G-action on N is admissible. For any subgroup H of G, let W^{H} denote the subgroup of W generated by the vertices of N which are fixed by H (i.e., W^{H} is the Coxeter group corresponding to the flag complex N^{H}). Then for any finite subgroup H of G we have the following:

• The fixed set of H on U(M, N, G) is U(M^{H}, N^{H}, V_{H}), where, as before, V_{H} = N_{G}(H)/H.

• W^{H} ⋊ V_{H} = N_{W ⋊G}(H)/H.

3. The equivariant Bestvina-Brady construction

We recall the construction of [1] and its extension in [12]. This time, suppose we are given as data:

• A finite flag complex L.

• A group Q of admissible symmetries of L. To this data we will associate a right-angled Artin group A_{L} and a CAT (0) space X_{L} on which A_{L} ⋊ Q is represented as a discrete, cocompact group of isometries (so X_{L} = \underline{E}G where G = A_{L} ⋊ Q). We will also have a subgroup B_{L} of A_{L} and a subspace Y_{L} of X_{L} so that B_{L} acts cocompactly on Y_{L}.

Let I denote the vertex set of L. The group A_{L} has generating set {a_{i}}_{i∈I} and a relation of the form [a_{i}, a_{j}] = 1, whenever {i, j} spans an edge of L. Let ϕ : A_{L} → ℤ be the homomorphism which sends each generator a_{i} to 1. The kernel of ϕ, denoted by B_{L}, is the Bestvina-Brady group associated to L. The Q-action on I induces a Q-action on A_{L} (through automorphisms) and the subgroup B_{L} is Q-stable. Hence, we can form the semidirect products A_{L} ⋊ Q and B_{L} ⋊ Q. The homomorphism ϕ extends to a homomorphism (also denoted by ϕ) from A_{L} ⋊ Q to ℤ by sending Q to 0. As in [6], A_{L} is a cocompact group of isometries CAT (0) cubical complex X_{L}. Since A_{L} is torsionfree, X_{L}/A_{L} = BA_{L}. The Q-action on L induces an isometric action on X_{L}/A_{L} and it is easily checked that the group of all lifts of this action to X_{L} is just A_{L} ⋊ Q. As in [1], there is a ϕ-equivariant map f : X_{L} → ℝ. In general, the homotopy type of the level set f^{−1}(t) will vary as t varies in ℝ; however, there is one important case when it does not: when L is contractible. In any case, fix a t ∈ ℝ and let Y_{L} denote the level set f^{−1}(t).

Assume that the Q-action on L is admissible. As explained in [12], in the above construction, the process of taking fixed sets of Q works as one might predict, i.e.,

(X_{L})^{Q} = X_{L^{Q}}
(Y_{L})^{Q} = Y_{L^{Q}}

and the subgroup of A_{L} fixed by Q is A_{L^{Q}}. If we regard Q as a subgroup of B_{L} ⋊Q and if V_{Q} stands for N_{B_{L}⋊Q}(Q)/Q, then V_{Q} = B_{L^{Q}} (cf., [12]).

The main result of [1] is that B_{L} is type F if and only if L is contractible. The main idea of [12] is that if L is contractible but the Q-action is such that L^{Q} is not contractible, then B_{L} ⋊ Q will not be type VF (since V_{Q} will not be type F). In addition, it is proved in [12] that if L^{Q} = ∅, then B_{L} ⋊ Q contains an infinite number of conjugacy classes of subgroups isomorphic to Q (all of which are conjugate in A_{L} ⋊ Q). Furthermore, there are many examples of such Q-actions on contractible finite simplicial complexes L; for example, see [3] and [16].

4. The first class of examples

Let Y_{L} be one of the examples of [12]. To simplify notation, set Ỹ = Y_{L}, Γ = B_{L} ⋊ Q and Y = Y_{L}/B_{L} = Ỹ/B_{L}. Thus, Y is a finite complex with fundamental group B_{L} and with universal cover Ỹ; the finite group Q acts cellularly on Y and the group of all lifts of the Q-action to Ỹ is Γ.

We want to Q-equivariantly “thicken” Y to a compact manifold M with boundary N. There are at least two methods for doing this. One is to use a theorem of Mostow [15] to Q-equivariantly embed Y in a linear representation of Q on some Euclidean space ℝ^{n}. (A proof of Mostow’s Theorem can be found in [3], pp. 110–112.) Then choose a Qequivariant triangulation of ℝ^{n} and let M be a regular neighborhood of Y in ℝ^{n}. Another method is to give Y the structure of a Q CW complex and then, as in [16], build the thickening M by using Q-handles attached by maps, which are Q-homotopic to the attaching maps for Y. In either case, we obtain a Q-action on M, a compact manifold with boundary, such that M equivariantly deformation retracts onto Y. Equivariantly triangulate the boundary N as a flag complex. Let M̃ denote the universal cover of M and let Ñ denote the inverse image of N in M̃ (give Ñ the induced triangulation). The group of all lifts of the Q-action to M̃ is Γ (since this is true for the Q-action on Y and Y is Q-homotopy equivalent to M.)

Now we apply the equivariant reflection group trick of Section 2. First consider the manifold U(M, N, Q). The group which acts on it is W ⋊ Q where W is the right-angled Coxeter group associated to N. The commutator subgroup T of W is a torsion-free subgroup of finite index. The manifold U(M̃, Ñ, Γ) is clearly a covering space of U(M, N, Q). It is contractible since M̃ is. The group which acts on U(M̃, Ñ, Γ) is W̃⋊Γ, where W̃ is right-angled Coxeter group associated to Ñ. The quotient space is M/Q which is compact. Let T̃ denote the inverse image of T in W̃. It is a torsion-free subgroup of finite index in W̃. So, T̃ ⋊ B_{L} is a torsion-free subgroup of finite index in W̃ ⋊ Γ. Thus, W̃ ⋊ Γ is type VF. This gives the following more precise version of Theorem 1. Theorem 3. The virtually torsion-free group W̃ ⋊ Γ acts properly and cocompactly on the contractible manifold U(M, N, Q). Moreover,

(1) If L^{Q} is empty, then W̃ ⋊ Γ contains infinitely many conjugacy classes of finite subgroups isomorphic to Q.

(2) If L^{Q} is nonempty and not contractible, then N_{W̃⋊Γ}(Q)/Q contains a torsion-free subgroup of finite index which retracts onto B_{L^{Q}} and hence is not type F

So, in either case, W̃ ⋊ Γ is not type VF.

Proof. If L^{Q} = ∅, then Γ contains infinitely conjugacy classes of subgroups isomorphic to Q and therefore, the same is true for the semidirect product W̃ ⋊ Γ. As explained in Section 2,

N_{W̃⋊Γ}(Q)/Q = W̃^{Q} ⋊ V_{Q}

and as explained in Section 3, V_{Q} = B_{L^{Q}}. Hence, for a suitable torsionfree subgroup T̃^{Q} of finite index in W̃^{Q}, the subgroup T̃^{Q}⋊B_{L^{Q}} retracts onto B_{L^{Q}}. □

5. The second class of examples

In this section, the finite group will act on a sphere rather than on a contractible complex. Again we use the equivariant reflection group trick of Section 2. Suppose that N is a triangulation of the (n − 1)- sphere as a flag complex and that a finite group G acts admissibly, by simplicial automorphisms on N. So, Cone(N) will be an n-disk. To simplify notation, set Σ = U(Cone(N), N, G). Then Σ is an ndimensional CAT (0) manifold equipped with an isometric action of W ⋊ G. As we noted in the Introduction, this implies that W ⋊ G is type VF.

For any finite subgroup H of G we have

Σ^{H} = U(Cone(N^{H}), N^{H}, V_{H})

where V_{H} = N_{G}(H)/H. One of the main results of [8] is a calculation of the cohomology with compact supports of such spaces. In particular, Theorem A of [8] yields

H_{c}^{i}(Σ^{H}) = ∑_{w∈W^{H}} \overline{H}^{i−1}(N^{H} − σ(w))

where σ(w) is a certain simplex of N^{H} depending on w ∈ W^{H}. It follows from this formula (cf., Theorem B in [8]) that the subgroup H of W ⋊ G satisfies the condition of the definition in Section 1 if and only if N^{H} is a “generalized homology sphere”, in that sense that it is a homology manifold with the same homology as a sphere. So, we have proved the following.

Theorem 4. The group W ⋊ G is a virtual Poincaré duality group. It is a VPD-group if and only if N^{H} is a generalized homology sphere for each finite subgroup H of G.

Since there are many examples of finite group actions on spheres where the fixed sets do not have the homology of a sphere (see [3]), Theorem 4 implies Theorem 2. We give a concrete example below. Example. Let N(a_{1}, …, a_{n}) denote the link of the origin in the complex hypersurface

(z_{1})^{a_{1}} + ⋯ + (z_{n})^{a_{n}} = 0.

It is a smooth manifold of dimension 2n − 3, called a “Brieskorn manifold”. As Brieskorn showed, it is often homeomorphic to S^{2n−3}, see [14]. For example, N = N(3, 2, 2, 2) is diffeomorphic to S^{5}. If ζ denotes a sixth rooty of unity, then define an action of ℤ/6 on N by the formula

ζ ▪ (z_{1}, z_{2}, z_{3}, z_{4}) = (ζ^{2}z_{1}, ζ^{3}z_{2}, z_{3}, z_{4}).

The fixed set of ℤ/3 on N is N(2, 2, 2) which is ℝP^{3}, the fixed set of ℤ/2 is N(3, 2, 2) which is a 3-dimensional lens space with fundamental group ℤ/3 and the fixed set of ℤ/6 is N(2, 2) which is the disjoint union of two circles. Equivariantly triangulate N as a flag complex and form the CAT (0) 6-manifold Σ as above. For any subgroup H of ℤ/6, let Ṽ_{H} = W^{H} ⋊ V_{H} be the group corresponding to H. For H = ℤ/3 or ℤ/2, Ṽ_{H} has cd_{ℚ} = 4 and H^{4}(Ṽ_{H}; ℤṼ_{H}) = ℤ. However, H^{3}(Ṽ_{H}; ℤṼ_{H}) is either an infinite sum of ℤ/2’s (when H = ℤ/3) or of ℤ/3’s (when H = ℤ/2). For H = ℤ/6, the group Ṽ_{H} is of cohomological dimension 2 and its second cohomology group is free abelian of infinite rank. So, W ⋊ G is not a VPD-group.

6. Appendix: type VF P

The orbit category, O(G), of a discrete group G is the category whose objects are the G-sets G/H for H any subgroup of G, with morphisms the G-maps between these G-sets. This is equivalent to a category whose objects are all transitive G-sets. The proper orbit category, O or O(G, F), is the full subcategory of O(G) whose objects are the transitive G-sets G/H for H finite. This is equivalent to a category whose objects are all transitive proper G-sets. A Bredon coefficient system for proper G-spaces or a ℤO-module is a contravariant functor from O to abelian groups, and a morphism of ℤO-modules is a natural transformation between two such functors. The category of ℤO-modules is an abelian category.

The trivial ℤO-module Z is defined to be the constant functor on O that sends every object to ℤ and every morphism to the identity map of ℤ. For each object G/H of O, define a ℤO-module P_{G/H} by

P_{G/H}(−) = ℤMap_{G}(−, G/H),

so that the value of P_{G/H} on G/K is the free abelian group with basis the G-maps from G/K to G/H. It may be shown that each P_{G/H} is projective, and that any ℤO-module is the homomorphic image of some direct sum of these modules. Call P_{G/H} the free ℤO-module of rank one based at G/H, and define a finitely generated free ℤOmodule to be a finite direct sum of modules P_{G/H}. A finitely generated projective ℤO-module is a direct summand of a finitely generated free ℤO-module. Definition. A group G is type VF P (resp. type VF L) if the trivial ℤO-module Z admits a finite resolution by finitely generated projective ℤO-modules (resp. finitely generated free ℤO-modules).

For a G-space X, the H-fixed points X^{H} may be identified with Map_{G}(G/H, X), and the map

G/H ↦ Map_{G}(G/H, X)

describes a contravariant functor from O to spaces. For a G CWcomplex, the map

G/H ↦ C_{∗}(Map_{G}(G/H, X))

describes a chain complex of free ℤO-modules, which are finitely generated if X contains finitely many orbits of cells. This construction yields a free resolution of Z when applied to \underline{E}G, and hence any group of type VF is of type VF L.

For more information concerning the category of ℤO-modules, see [13], which contains proofs of all the statements made in this section.

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Department of Mathematics, The Ohio State University, 231 W. 18th Avenue, Columbus, Ohio 43210, United States

E-mail address: mdavis@math.ohio-state.edu

Faculty of Mathematical Studies, University of Southampton, Southampton, SO17 1BJ, United Kingdom

E-mail address: I.J.Leary@maths.soton.ac.uk