On the classifying space for the family of virtually cyclic subgroups for elementary amenable groups

We show that elementary amenable groups, which have a bound on the orders of their finite subgroups, admit a finite dimensional model for the classifying space with virtually cyclic isotropy.


INTRODUCTION
Classifying spaces with isotropy in a family have been studied for a while; most of the research has focussed on EG, the classifying space with finite isotropy [19][20][21]. Finiteness conditions for EG for elementary amenable groups are very well understood [8,14]. Finding manageable models for EG has been shown to be much more elusive. In [12] it was conjectured that the only groups admitting a finite type model for EG are virtually cyclic, and this was proved for hyperbolic groups. In [13] it was shown that this conjecture also holds for elementary amenable groups. As far as finite dimensional models are concerned, only a little more is known. So far manageable models have been found for crystallographic groups [17], polycyclicby-finite groups [24], hyperbolic groups [12] and CAT(0)-groups [7,22]. Adapting the construction of [12], the first author [10] has recently found a good model in the case when G is a certain type of HNN-extension, including extensions of the form G = A ⋊ Z, where the generator of Z acts freely on the non-trivial elements of an abelian group A. Utilising this construction we prove:

Main Theorem. Let G be an elementary amenable group with finite Hirsch length. If G has a bound on the orders of its finite subgroups, then G admits a finite dimensional model for EG.
The proof of this fact is algebraic, and uses finiteness conditions in Bredon cohomology. Bredon cohomology takes the place of ordinary cohomology when studying classifying spaces with isotropy in a family of subgroups. We give a brief introduction into Bredon cohomology in Section 2 and then move on to discussing dimensions in Bredon cohomology for extensions, directed unions and direct products of groups. We also consider the behaviour of the Bredon cohomological dimension when changing the family of subgroups. A further crucial ingredient is Hillman-Linnell's and Wehrfritz' [11,34] characterisation of elementary amenable groups as locally finite-by-soluble-by-finite groups. This allows us to reduce the problem to torsion-free abelian-by-cyclic groups. We show, Proposition 5.4, that a torsion-free abelian-by-cyclic group of finite Hirsch length has finite Bredon-cohomological dimension bounded by a recursively defined integer only depending on the Hirsch length. In Section 6 this result is extended to torsion-free nilpotent-by-abelian groups, which allows us to prove the Main Theorem in Section 7.

BACKGROUND ON BREDON COHOMOLOGY
In this article a family F of subgroups of a group G stands for a non-empty set of subgroups of G, which is closed under conjugation and taking finite intersections. Common examples are the trivial family of subgroups F = {1}, the family F fin (G) of all finite subgroups of G and the family F vc (G) of all virtually cyclic subgroups of G. Let F be a family of subgroups of G and K ≤ G and put which is a family of subgroups of K. Now let F 1 and F 2 be families of subgroups of some groups G 1 and G 2 respectively. Here we put This is a family of subgroups of G 1 × G 2 . Finally, for any family F of subgroups of G we can define its subgroup completionF as That is,F is the smallest family of subgroups of G which contains F and is closed under forming subgroups. Given a non-empty G-set X , we denote by F(X ) the collection of all its isotropy groups. In general this is not a family of subgroups as it may not be closed under finite intersections.
Bredon cohomology was introduced for finite groups by Bredon [2] and it has been generalised to arbitrary discrete groups by Lück [19]. It is the natural choice for a cohomology theory to study classifying spaces with stabilisers in a prescribed family F of subgroups. The reader is referred to Lück's book [19] and the introductory chapters in Mislin's survey [27] for standard facts and definitions. We shall, however include those definitions and results on Bredon cohomology needed later on.  Given a covariant functor F: O F 1 G 1 → O F 2 G 2 between orbit categories, one can now define induction and restriction functors along F, see [19, p. 166]: and Res F : Proof. This follows from the fact that both Ind F and Tor F 1 (?, N) commute with filtered colimits and from Proposition 2.1.
Let F ⊂ G are two families of subgroups of a group G, then the inclusion of the respective orbit categories is denoted by: Note that for every non-empty K-sets X with F(X ) ⊂ F ∩ K the functor I K can be extended by mapping each K-orbit separately.

Lemma 2.3. [32, Lemma 2.9] Let K be a subgroup of H such that F ∩ K is a non-empty subset of F. Then induction with I K is an exact functor.
We conclude this section with a collection of facts concerning dimensions both generally and for the family of virtually cyclic subgroups. The Bredon cohomological dimension cd F G of a group G with respect to the family F of subgroups is the pro- we shall use the following notation: cd G = cd F G and hd G = hd F G. The cellular chain complex of a model for E F G yields a free resolution of the trivial O F G-module Z [19, pp. 151f.]. In particular, this implies that the Bredon geometric dimension gd F G, the minimal dimension of a model for E F G, is an upper bound for cd F G. Since projectives are flat this implies that Furthermore, Lück and Meintrupp gave an upper bound of gd F G in terms of cd F G as follows; the case F = F fin (G) was shown in [19]: Hence, as long as cd F G ≥ 3 or gd F G ≥ 4, we have equality of these two dimensions.
Now suppose H is a subgroup of G such that F ∩ H is a non-empty subset of F. Then gd F∩H H ≤ gd F G and cd F∩H H ≤ cd F G. The following result is a consequence of Martínez-Pérez' Lyndon-Hochschild-Serre spectral sequence in Bredon (co)homology [26]. We shall only state the results for the family of virtually cyclic subgroups.
A careful inspection of the terms of the spectral sequence [26,Theorem 4.3] yields the following: Proof. The proof is identical to that for the corresponding result for the family of all finite subgroups [28,Theorem 5.5]. One checks that the families in question satisfy the conditions of [26,Corollary 4.5]. Now suppose G is a finite extension of a group H. Lück has constructed a model for EG from a model for EH [20]. This yields the following bound for gd G:

In particular, gd G is finite if and only if gd H is finite.
In light of Proposition 2.5 one needs to understand the behaviour of the Bredon dimensions for the family of virtually cyclic subgroups under extensions with virtually cyclic quotients. In [10] the first author gave bounds for certain infinite cyclic extensions:

DIRECTED UNIONS OF GROUPS
The standard resolution of Z in classical group cohomology [3, pp. 15f] has been extended to Bredon cohomology for the family F fin (G) of all finite subgroups of a given group G [28]. This construction can be generalised to arbitrary families F without any essential changes: Since F is closed under taking finite intersections it follows that where (g 0 K 0 , . . . , g i K i , . . . , g n K n ) denotes the n-tuple obtained from the (n + 1)tuple (g 0 K 0 , . . . , g n K n ) by deleting the i-th component.
We now obtain a resolution of the trivial Since F(∆ n ) ⊂ F it follows that this resolution is free and it is called the standard resolution of the trivial O F G-module Z.
There now follows a variation of [28,Theorem 4.2].
Note that Part (ii) has also been derived in [5,Corollary 4.3], using a spectral sequence argument.
Proof. This follows from the fact that F vc (G λ ) = F vc (G) ∩ G λ and that for every finitely generated subgroup H there is a λ ∈ Λ such that H ∈ G λ . Now apply Proposition 3.1.
In particular, Corollary 3.2 (ii) can be applied to countable groups. A countable group is the direct union of its finitely generated subgroups G λ , λ ∈ Λ, where Λ is countable. Hence cd G ≤ sup{cd G λ } + 1.
Before we can prove Proposition 3.1, we need the following technical lemma.

Lemma 3.3.
(i) Assume that the G-set X is the direct union of G-sets X α . Then the homomorphism induced by the canonical inclusions Z[?, induced by the projections G/G λ ։ G/G is an isomorphism.
and this isomorphism is natural in X .
Proof. (i) Let H ∈ F and evaluate (1)  (ii) This follows directly from the universal property of a colimit.
(iii) Let R be a complete system of representatives of the orbit space X /K. Then we have the following sequence of isomorphisms of O F G-modules: Note that the third isomorphism is a consequence of the Yoneda-Lemma and that the composition of these isomorphisms is clearly natural in X .

Proof of Proposition
By Lemma 2.3 the functor Ind I G λ is exact. Hence for each λ ∈ Λ there is an exact sequence of O F G-modules: where X λ ,n = I G λ (∆ λ ,n ). Note that the X λ ,n are G-invariant subsets of ∆ n and that ∆ n is the directed union of the X λ ,n . For each λ ≤ µ the inclusion X λ ,n ֒→ X µ,n , n ≥ 0, induces a homomorphism η µ λ ,n : Z[?, X λ ,n ] G → Z[?, X µ,n ] G . Also, the projection G/G λ ։ G/G µ induces homomorphisms η µ λ ,−1 : Hence we have chain-maps between the corresponding chain complexes (5). These chain complexes together with the chain maps η µ λ , * form a direct limit system indexed by Λ. Lemma 3.3 (i) and (ii) imply that its limit is the sequence Since direct limits preserve exactness [35, p. 57], this sequence is exact.
Denote by K λ ,n the n-th kernel of the sequence (4). As before, Ind G λ (K λ ,n ) is the n-th kernel in (5) and the chain maps η µ λ ,n yield a inverse limit system. Since taking direct limits preserves exactness we get that is the n-th kernel of (6). Now suppose that there exists a n ∈ N such that hd F λ G λ ≤ n for all λ ∈ Λ. In particular, all K λ ,n are flat. Now Lemma 2.2 implies that Ind G λ K λ ,n all are flat. Since Tor F 1 (?, M) commutes with direct limits it follows that K n is a flat O F G-module. In particular, hd F G ≤ n proving (i). The proof of (ii) is analogous. Apply [28,Lemma 3.4], which states that a countable colimit of projective O F G-modules has projective dimension ≤ 1.

Lemma 3.4. Let A be a countable abelian group with finite Hirsch length h(A).
Then Proof. Write A as the countable direct union A = lim − → A λ of its finitely generated subgroups A λ . [24, Theorem 5.13] implies gd A λ ≤ h(A λ ) + 1, and hence cd A λ ≤ h(A λ ) + 1 ≤ h(A) + 1. Thus, by Corollary 3.2, cd A ≤ h(A) + 2 as required.

CHANGE OF FAMILY AND DIRECT PRODUCTS OF GROUPS
The following result is the algebraic counterpart to [24, Proposition 5.1 (i)]. Although we only state and prove it for Bredon cohomology, an analogous statement also holds for Bredon homology. The result for Bredon cohomology has also been proved in [5, Corollary 4.1] using a spectral sequence argument.  Let G 1 and G 2 be groups and let F 1 and F 2 be subgroup-closed families of subgroups of G 1 and G 2 respectively. Let G = G 1 × G 2 and F = F 1 × F 2 and take G ⊂F to be a subgroup-closed family of subgroups of G. Assume that there exists k ∈ N such that cd G∩K K ≤ k for every K ∈ F. Then

Proposition 4.1. Let G be a group and F and G two families of subgroups of G such that F ⊂ G and that, for every K
Similar results for the families F 1 = F 2 = F fin and G-CW-complexes have been obtained in [18,30]. Also note that the natural projections p i : G → G i give rise to functors

Proof of Proposition 4.2.
Let P * ։ Z F 1 and Q * ։ Z F 2 be free resolutions. Hence there exist G 1 -sets X i and G 2 -sets Y j such that P i = Z[?, Let P ′ * = Res p 1 P * and Q ′ * = Res p 2 Q * . These O F G-modules are of the form where the action of G on X i and Y i is given by gx = p 1 (g)x and gy = p 2 (g)y respectively. For each i, j ∈ N we have an identification of O F G-modules Here G acts diagonally on X i ×Y j . This gives rise to a double complex in the usual way, see for example [35, pp. 58f.]. Denote by C k its total complex: The augmentation maps ε 1 : P 0 ։ Z F 1 and ε 2 : Q 0 ։ Z F 1 induce an augmentation map ε: C 0 ։ Z F . Altogether we obtain a resolution Now the free O F G-modules C k are also free OFG-modules. Since the families F 1 and F 2 are assumed to be subgroup closed we can extend, using Remark 4.5, every morphism in the sequence (8) to a morphism of the corresponding OFG-modules. It follows that we obtain a resolution of the trivial OFG-module by free OFG-modules. Now assume that m = cd F 1 G 1 and n = cd F 2 G 2 . Then it follows from an Eilenberg Swindle that there are free resolutions P * ։ Z F 1 and Q * ։ Z F 2 as above, of lengths m and n respectively. This implies that C k = 0 for all k > m + n. In particular Let K ∈F. Then K ≤ K 1 × K 2 for some K 1 × K 2 ∈ F. Since G is assumed to be closed under forming subgroups it follows that ∅ = G ∩ K ⊂ G ∩ (K 1 × K 2 ). Therefore we have cd G∩K K ≤ cd G∩(K 1 ×K 2 ) (K 1 × K 2 ). By assumption the latter is bounded by k. Thus we have cd G G ≤ cdF G + k by Proposition 4.1 and the claim of the proposition follows.
Remark 4.6. Note that the special case of Corollary 4.3 follows almost immediately by applying Martínez-Pérez' spectral sequence Proposition 2.5 twice, but we have included the above for its generality and for being rather elementary. The alternative argument is as follows: Consider G = G 1 × G 2 as an extension This gives rise to an extension Applying Proposition 2.5 again yields that cd H ≤ n + cd G 1 where n is the supremum of cd L where L ranges over all L ≤ H with V ≤ L and L/V is virtually cyclic. These L are of the form V ×W with W a virtually cyclic subgroup of G 1 . Thus where k is the supremum of cd(V 1 × V 2 ) with V 1 and V 2 ranges over all virtually cyclic subgroups of G 1 and G 2 respectively. In particular k ≤ 3 by [24, Theorem 5.13].

Lemma 5.1. Let G be a torsion-free abelian-by-(infinite cyclic) group, i.e. there is a short exact sequence
A G ։ t with A abelian and t ∼ = Z. Consider the subgroupĀ = {a ∈ A | a t = a}. Then G/Ā is torsion-free.
Proof.Ā is obviously a central subgroup in G and we have a short exact sequence To prove the claim it suffices to show that A/Ā is torsion-free. Suppose there is an a ∈ A such that π(a) n = π(a n ) = 1. This implies that a n ∈Ā and hence (a n ) t = a n . Since A is abelian, we have (a t a −1 ) n = 1. But A is torsion-free and hence a t = a. This implies a ∈Ā.
In a torsion-free abelian-by-(infinite cyclic) group, the generator t of the infinite cyclic group acts by automorphisms on the abelian group A. As we will see in Proposition 5.4, there is no problem if t acts trivially or freely on the non-trivial elements of A. The main problem arises when t acts by a finite order automorphism. But the following, folklore, version of Selberg's Lemma tells us that the order of this automorphism has a bound only depending on A. We shall state the Lemma as a special case of [ For our purpose we need the following consequence of Selberg's Lemma, which is probably known. We include it for completeness.

Lemma 5.3. Let A be a torsion-free abelian group with finite Hirsch length h(A). Then there exists a integer ν = ν(A) which depends only on h(A) such that for any automorphism t of A the finite orbits of elements in A under the action of t have at most length ν.
Proof. Let n = h(A). Then A ⊗ Q ∼ = Q n and A can be viewed as an additive subgroup of Q n by a → a ⊗ 1. The automorphism t of A extends to an automorphism t ⊗ id : Q n → Q n of Q-vector spaces, which we denote by ϕ.
It has a complement V in Q n and there exists a unique linear map ψ: Q n → Q n which agrees with ϕ on U and which is the identity on V . Then ψ is an isomorphism which is periodic by construction. By Selberg's Lemma there exists a number ν(n) such that every periodic automorphism of Q n has order at most ν(n). Therefore ψ m = id for some 1 ≤ m ≤ ν(n).
In particular we have that ϕ m (a) = a for each a ∈ U ∩ A.

Proposition 5.4. Let A be a torsion-free abelian group of finite Hirsch length h.
There is a recursively defined integer f (h) depending only on h such that for every infinite cyclic extension G = A ⋊ t we have Proof. We prove the proposition by induction on the Hirsch length of A. Since A is torsion-free, h = 0 implies that A is trivial. In this case G is infinite cyclic and therefore f (0) = 0. Now suppose h ≥ 1 and assume that the statement is true for all torsion-free abelian groups B with h(B) < h. Let A be a torsion-free abelian group with Hirsch length h and letĀ be as in Lemma 5.1. Then precisely one of the following three cases occurres. (1) (2) {1} =Ā but there is an element 1 = a ∈ A and a positive integer m such that a t m = a: Lemma 5.3 implies that m is bounded by a number ν that only depends on h. Let r h = lcm(1, . . . , ν) and set A 0 = A and t 0 = t r h .
(Case (1)) (3) {1} =Ā and for all 1 = a ∈ A and all m = 0 we have a t m = a: Then Proposition 2.8 applies to G and it follows that gd G ≤ gd A + 1. Thus where the last inequality is due to Theorems 4.3 and 5.13 in [24].

NILPOTENT-BY-ABELIAN GROUPS
For any group G we denote its centre by Z(G).

PROOF OF THE MAIN THEOREM
The proof is now an easy application of a theorem by Hillman and Linnell [11]. We shall refer to an alternative proof of their theorem, see points (f) and (g) Wehrfritz [34], whose statement is better suited to our purpose. For any group G we denote by τ(G) its unique maximal normal locally finite subgroup. Proof of the Main Theorem. Since G has a bound on the orders of the finite subgroups, τ(G) is finite. An application of Proposition 2.6 allows us to assume that τ(G) = {1} and hence that G is virtually torsion-free. Using Proposition 2.7 we can assume that G is torsion-free nilpotent-by-abelian. Hence we can apply Theorem 6.2.
Remark 7.2. To remove the condition that there is a bound on the orders of the finite subgroups, one needs to understand virtually cyclic extensions of large locally finite groups. This would allow us to apply Martínez-Pérez' spectral sequence as before. Since τ(G) is locally finite, every virtually cyclic subgroup is, in fact, finite and hence E(τ(G)) = E(τ(G)) and these are well understood [6]. In a recent article Degrijse and Petrosyan have provided bounds for the dimension of ET for T locally finite-by-virtually cyclic [4]. This implies that every elementary amenable group G admits a finite dimensional model for EG.