GEOMETRIC DIMENSION OF GROUPS FOR THE FAMILY OF VIRTUALLY CYCLIC SUBGROUPS

DIETER DEGRIJSE AND NANSEN PETROSYAN

Abstract. By studying commensurators of virtually cyclic groups, we prove that every elementary amenable group of finite Hirsch length h and cardinality ℵ_{n} admits a finite dimensional classifying space with virtually cyclic stabilizers of dimension n + h + 2. We also provide a criterion for groups that fit into an extension with torsion-free quotient to admit a finite dimensional classifying space with virtually cyclic stabilizers. Finally, we exhibit examples of integral linear groups of type F whose geometric dimension for the family of virtually cyclic subgroups is finite but arbitrarily larger than the geometric dimension for proper actions. This answers a question posed by W. Lück.

Contents

1. Introduction 1 2. Preliminaries and Notations 4 3. Commensurators of infinite virtually cyclic groups 8 4. Elementary amenable groups 10 5. Examples 14 6. Extensions with torsion-free quotient 20 7. The Mayer-Vietoris sequence of Lück and Weiermann 28 References 35

1. Introduction

A classifying space of a discrete group G for a family of subgroups F is a terminal object in the homotopy category of G-CW complexes with stabilizers in F (see [38]). Such a space is also called a model for E_{F}G. A model for E_{F}G always exists for any given discrete group G and a family of subgroups F but it need not be of finite type or finite dimensional. When F is the family of virtually cyclic subgroups, E_{F}G is denoted by \underline{\underline{E}}G. Questions concerning finiteness properties of \underline{\underline{E}}G have been especially motivated by the Farrell-Jones Isomorphism Conjecture which describes the structure of the algebraic K- and L-theory of the group ring of G by a certain equivariant homology theory of \underline{\underline{E}}G (e.g. see [1],[4] and [10]).

Both authors were supported by the Research Fund KU Leuven.

The second author was also supported by the FWO-Flanders Research Fellowship.

Finite dimensional models for \underline{\underline{E}}G have been constructed for several interesting classes of groups, for example, word-hyperbolic groups (Juan-Pineda, Leary, [19]), relatively hyperbolic groups (Lafont, Ortiz, [21]), virtually polycyclic groups (Lück,Weiermann, [29]) and CAT(0)-groups (Farley, [9], Lück, [27]).

In [5], we have shown that every elementary amenable group G of finite Hirsch length and cardinality ℵ_{n} admits a finite dimensional model for \underline{\underline{E}}G. However, an explicit upper bound for the minimal dimension of a model for \underline{\underline{E}}G was not given. In Section 4, we prove the following.

Theorem A. Let G be an elementary amenable group of finite Hirsch length h and cardinality ℵ_{n}. Then there exists a model for \underline{\underline{E}}G of dimension n + h + 2.

Let us point that if an elementary amenable group G has infinite Hirsch length then it cannot admit a finite dimensional model for \underline{\underline{E}}G. In Section 5, we give examples of groups that show that the bounds obtained in the theorem are attained for each positive value of the Hirsch length h and any cardinality ℵ_{n}. Theorem A thus answers problems 46.1 and 46.2 stated by Lafont in [20] for the class of elementary amenable groups of cardinality ℵ_{n}.

The smallest possible dimension of a model for \underline{\underline{E}}G is called the geometric dimension of G for the family of virtually cyclic subgroups, denoted by \underline{\underline{gd}}(G). In Section 6, we address the following basic question.

Question. If Γ fits into a short exact sequence of groups N → Γ → Q such that \underline{\underline{gd}}(N) and \underline{\underline{gd}}(Q) are both finite, is \underline{\underline{gd}}(Γ) also finite?

The answer to this question and the analogous question for the family of finite subgroups is unknown in general (see [32], [26, 5.5]). We give a positive answer provided that the group Q is torsion-free and the number δ(N) defined below is finite.

For a given group N of cardinality ℵ_{n}, we let E_{N} be the collection of all groups that can be formed as an extension of a finitely generated subgroup of N by a finite cyclic group. We then define

δ(N) = sup\{\underline{gd}(G) |G ∈ E_{N}\} + n,

where \underline{gd}(G) is the geometric dimension of G for proper actions.

Theorem B. Let N be a group of cardinality ℵ_{n} and let N → Γ → Q be a short exact sequence of groups such that Q is torsion-free. Then

\underline{\underline{gd}}(Γ) ≤ \underline{\underline{gd}}(Q) + max\{\underline{\underline{gd}}(N), δ(N)\} + 2.

There are several types of groups N for which δ(N) is bounded by \underline{gd}(N) + 1 and it is a general fact that \underline{gd}(N) is at most as large as \underline{\underline{gd}}(N) + 1. For example, if N is a group with \underline{gd}(N) = 1, such as a virtually free group, then δ(N) ≤ 1 (see [8, 1.1-2]). Also, if N is a finite extension of an orientable surface group, then δ(N) = \underline{gd}(N) = 2 (see [33, 4.4(2)]). Lastly, if N is a countable elementary amenable group of finite Hirsch length h(N), then δ(N) ≤ h(N) + 1 and h(N) ≤ \underline{gd}(N) (see [11]).

There are also groups N that have δ(N) greater than \underline{gd}(N) + 1. In fact, in Example 6.5, we show that there are groups for which δ(N) is finite but arbitrarily larger than \underline{gd}(N). A general method of constructing such groups is given in Theorem C. Let us note that we are not aware of an example of a group N for which \underline{gd}(N) is finite but δ(N) is infinite. In Remark 6.6, we point out how an example of a group with this property may be used to construct a 2-generated group satisfying \underline{gd}(Γ) < ∞ and \underline{\underline{gd}}(Γ) = ∞, thus answering a question of Lafont in [20, 46.4].

Recall that if X is a property of a group, then a group is said to be of type V X if it contains a finite index subgroup of type X. Given a family of subgroups F of a group G, one can consider the Bredon cohomological dimension cd_{F}(G) of the group G for the family F (see Section 2), which satisfies

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

When F is the family of finite or virtually cyclic subgroups of the group G, then cd_{F}G is denoted by \underline{cd}G or \underline{\underline{cd}}G, respectively.

Theorem C. Let {L_{i} | i = 1, …, s} be a collection of finite acyclic flag complexes. Let {p_{i} | i = 1, …, s} be a collection of primes and q be a prime that is not in this set. For each i, suppose the complex L_{i} is of dimension n_{i} ≥ 1 and the cyclic group C_{p_{i}} acts admissibly and simplicially on L_{i} such that

H_{n_{i}−1}(L_{i}^{C_{p_{i}}}, 𝔽_{q}) ≠ 0.

For each i, denote by H_{i} the Bestvina-Brady group associated to L_{i}. Let H_{i} ⋊ C_{p_{i}} be the semi-direct product formed by the automorphism action of C_{p_{i}} on H_{i} that is induced by the action of C_{p_{i}} on L_{i}.

(a) Let Γ_{s} = ∏_{i=1}^{s} H_{L_{i}} ⋊ C_{p_{i}}. Then Γ_{s} is an integral linear group of type V F P such that

vhd(Γ_{s}) = vcd(Γ_{s}) = ∑_{i=1}^{s} n_{i} and \underline{hd}(Γ_{s}) = \underline{cd}(Γ_{s}) = s + ∑_{i=1}^{s} n_{i}.

(b) Suppose further that s ≥ 2 and p_{i} ≠ p_{j} for all 1 ≤ i < j ≤ s. Let Λ_{s} = (∏_{i=1}^{s} H_{L_{i}}) ⋊ ℤ be the pull-back of Γ_{s} under projection of the infinite cyclic group ℤ onto ∏_{i=1}^{s} C_{p_{i}}. Then Λ_{s} is an integral linear group of type F P such that

hd(Λ_{s}) = cd(Λ_{s}) = 1 + ∑_{i=1}^{s} n_{i} and s + ∑_{i=1}^{s} n_{i} ≤ \underline{\underline{cd}}(Λ_{s}) < ∞.

Moreover, if L_{i} is contractible for each i = 1, …, s, then Γ_{s} is of type V F and Λ_{s} is of type F.

A question posed by Lück in [27] (see also [29]) asks whether or not the inequality \underline{\underline{gd}}(Γ) ≤ \underline{gd}(Γ) + 1 holds for every group Γ. In Corollary 4.4, we prove that all countable elementary amenable groups of finite Hirsch length satisfy this inequality. On the other hand, in Example 6.5 we apply Theorem C(b) to construct groups that do not satisfy this inequality.

We do not claim originality for the statement on the Bredon cohomological dimension of Γ_{s} obtained in Theorem C(a) in the case of s = 1. This result follows from a theorem of Martínez-Pérez (see [31, 3.3]), which led us to the construction of the groups Γ_{s}. However, we note that our methods for computing the Bredon cohomological dimension of these groups differ from those of [31].

In many computations, we use a Mayer-Vietoris long exact sequence of Lück and Weiermann which can be constructed using a certain push-out of classifying spaces. In the last section, we give a purely algebraic proof of the existence of this sequence which has the added benefit that it can be generalized to Tor and Ext functors.

2. Preliminaries and Notations

Let G be a discrete group and let F be a family of subgroups of G, i.e. a collection of subgroups of G that is closed under conjugation and taking subgroups. A classifying space of G for the family F is a G-CW-complex X such that X^{H} = ∅ when H ∉ F and X^{H} is contractible for every H ∈ F. A classifying space of G for the family F is also called a model for E_{F}G. It can be shown such a model always exists and is unique up to G-homotopy. The smallest possible dimension of a model for E_{F}G is called the geometric dimension of G for the family F and is denoted by gd_{F}(G). When a finite dimensional model does not exist, then gd_{F}(G) is said to be infinite. When F is the family of finite (virtually cyclic) subgroups, then E_{F}G and gd_{F}G are denoted by \underline{E}G (\underline{\underline{E}}G) and \underline{gd}G (\underline{\underline{gd}}G), respectively.

We will now state some known results on the finiteness of gd_{F}(G) that will be used throughout this paper, some without explicit referral.

The first result gives an upper bound for gd_{F}(G), when F is either the family of subgroups of finitely generated subgroups or the family of countable subgroups of G.

Theorem 2.1. Let G be a group of cardinality ℵ_{n}, for some natural number n, let F be the family of subgroups of finitely generated subgroups of G and let H be the family of countable subgroups of G. Then there exists a model for E_{F}G of dimension n + 1, and a model for E_{H}G of dimension n.

Proof. The statement for the family F is exactly Theorem 5.31 in Lück-Weiermann, [29, 5.31]. The proof of this theorem can be easily adapted to deal with the family of countable subgroups H. □

Let G be a group and let F and H be two families of subgroups of G such that F ⊆ H. Next, we will deal with the following questions.

(a) How can a model for E_{F}G be obtained from a model for E_{H}G?

(b) How can a model for E_{H}G be obtained from a model for E_{F}G?

For a subgroup H of G, one can define the family

F ∩ H = {H ∩ F | F ∈ F}

of subgroups of H. By restricting the action of G to H, any model for E_{F}G becomes a model for E_{F∩H}H. This implies that gd_{F∩H}(H) ≤ gd_{F}(G).

The following theorem gives us an answer to (a).

Theorem 2.2 (Lück-Weiermann, [29, 5.1]). Let G be a group and let F and H be two families of subgroups of G such that F ⊆ H. If there is an integer k ≥ 0 such that for each H ∈ H, there is a k-dimensional model for E_{F∩H}H. Then there exists a model for E_{F}G of dimension k + gd_{H}(G).

Note that the theorem implies that \underline{gd}(G) ≤ \underline{\underline{gd}}(G) + 1, for any group G.

The next corollaries will be key tools in our future arguments.

Corollary 2.3. Consider a short exact sequence of groups N → G ⟶^{π} Q. Let F be a family of subgroups of G and let H be a family of subgroups of Q such that π(F) ⊆ H. If there is an integer k ≥ 0 such that for each H ∈ H, there is a k-dimensional model for E_{F∩π^{−1}(H)}π^{−1}(H). Then there exists a model for E_{F}G of dimension k + gd_{H}(Q).

Proof. Define the family

C = {K ⊆ G | K ⊆ π^{−1}(F) for some F ∈ H}

of subgroups of G. Note that F ⊆ C. Let X be a model for E_{H}Q. By letting G act on X via π, it follows that X is also a model for E_{C}G. This implies that gd_{C}(G) ≤ gd_{H}(Q). The statement now follows from Theorem 2.2. □

Corollary 2.4. Let G be a group of cardinality ℵ_{n} and let F be a family of subgroups.

(i) If there is an integer d ≥ 0 such that for every finitely generated subgroup H of G, there is a d-dimensional model for E_{F∩H}H. Then there exists a model for E_{F}G of dimension d + n + 1.

(ii) If there is an integer d ≥ 0 such that for every countable subgroup H of G, there is a d-dimensional model for E_{F∩H}H. Then there exists a model for E_{F}G of dimension d + n.

Proof. This follows from combining Theorem 2.1 and Theorem 2.2. □

One way of dealing with the question (b) is to start with a model for E_{F}G and then try to adapt this model to obtain a model for E_{H}G. In section 2 of [29], Lück and Weiermann give a general construction that follows this principle. We recall some of the basics of this construction that will be useful in the subsequent sections.

Let G be a discrete group and let F and H be families of subgroups of G such that F ⊆ H and such that there exists an equivalence relation ∼ on the set S = H ∖ F that satisfies the following properties

- ∀H, K ∈ S : H ⊆ K ⇒ H ∼ K;
- ∀H, K ∈ S, ∀x ∈ G : H ∼ K ⇔ H^{x} ∼ K^{x}.

An equivalence relation that satisfies these properties is called a strong equivalence relation. Let [H] be an equivalence class represented by H ∈ S and denote the set of equivalence classes by [S]. The group G acts on [S] via conjugation, and the stabilizer group of an equivalence class [H] is

(1)  N_{G}[H] = {x ∈ Γ | H^{x} ∼ H}.

Note that N_{G}[H] contains H as a subgroup. Define for each [H] ∈ I the family

(2)  F[H] = {K ⊂ N_{G}[H] |K ∈ S, K ∼ H} ∪ (N_{G}[H] ∩ F)

of subgroups of N_{G}[H].

Theorem 2.5 (Lück-Weiermann, [29, 2.5]). Let F ⊆ H be two families of subgroups of a group G such that S = H ∖ F is equipped with a strong equivalence relation. Denote the set of equivalence classes by [S] and let I be a complete set of representatives [H] of the orbits of the conjugation action of G on [S]. If there exists a natural number k such that gd_{F∩N_{G}[H]}(N_{G}[H]) ≤ k − 1 and gd_{F[H]}(N_{G}[H]) ≤ k for each [H] ∈ I, and such that gd_{F}(G) ≤ k, then gd_{H}(G) ≤ k.

Lück and Weiermann applied this construction to prove the following result which will be useful for our purposes.

Theorem 2.6 (Lück, [26, 5.26] and Lück-Weiermann, [29, 5.13]). Let Γ be a virtually polycyclic group of Hirsch length h, then \underline{gd}(Γ) = h and \underline{\underline{gd}}(Γ) ≤ h + 1.

An important algebraic tool to study finiteness properties of classifying spaces for families of subgroups is Bredon cohomology. Bredon cohomology was introduced by Bredon in [3] for finite groups and was generalized to arbitrary groups by Lück (see [24]).

Let G be a discrete group and let F be a family of subgroups of G. The orbit category O_{F}G is a category defined by the objects that are the left cosets G/H for all H ∈ F and the morphisms are all G-equivariant maps between the objects. 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 and is defined by the objects that are all the O_{F}G-modules and the morphisms are all the natural transformations between the objects. The category Mod-O_{F}G contains enough projective and injective objects to construct projective and injective resolutions, 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. 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 (virtually cyclic) subgroups, then cd_{F}G is denoted by \underline{cd}G (\underline{\underline{cd}}G). 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). In fact, in [28, 0.1], Lück and Meintrup even showed that

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

Also note that for any group G, one has cd_{ℚ}(G) ≤ \underline{cd}(G).

In the last section, we will briefly discuss some of the other technical aspects of Bredon cohomology. We will also recall the definition of Bredon homology and Bredon homological dimension, as we will need these notions in Section 6. We refer the reader to [24] and [13] for a detailed introduction to Bredon (co)homology.

Notation 2.7.

- If a_{1}, a_{2}, …, a_{n} are elements of some group G, then ∏_{i=1}^{n} a_{i} denotes the product a_{1}a_{2} … a_{n} in that specific order.

- By ℕ_{1} we denote the set of natural numbers without the number zero.

- If t is an element of some group G, then ⟨t⟩ is the cyclic subgroup of G generated by t. All infinite cyclic groups will be written multiplicatively with unit 1. The unit element of a group that is not infinite cyclic will be denoted by e.

- If G is a group and C = ⟨t⟩ is an infinite cyclic group generated by t, we say that a semi-direct product G ⋊ C is determined by φ ∈ Aut(G) when

(e, t)(g, 1)(e, t^{−1}) = (φ(g), 1) ∈ G ⋊ C

for all g ∈ G and denote this group by G ⋊_{φ} C.

- For any d ∈ ℕ_{1}, we will denote by C_{d} the cyclic group of order d.

- Unless stated otherwise, F will always denote the family of finite subgroups of a given group.

3. Commensurators of infinite virtually cyclic groups

In this section, we apply the construction of Lück and Weiermann when F and H are the families of finite and virtually cyclic subgroups of G, respectively. This naturally leads us to consider commensurators of infinite virtually cyclic subgroups of G. We derive some general facts about such commensurators that will be essential to the proof of Theorem A.

The commensurator of a subgroup H of a group G is by definition the group

Comm_{G}[H] = {x ∈ G | [H : H ∩ H^{x}] < ∞ and [H^{x} : H ∩ H^{x}] < ∞}.

When H is an infinite virtually cyclic subgroup of G then one can verify that

Comm_{G}[H] = {x ∈ G | |H ∩ H^{x}| = ∞}.

Let us denote the set of infinite virtually cyclic subgroups of G by S. As in definition 2.2. of [29], two infinite virtually cyclic subgroups H and K of Γ are said to be equivalent, denoted H ∼ K, if |H ∩ K| = ∞. Using the fact that any two infinite virtually cyclic subgroups of a virtually cyclic group are equivalent (e.g. see Lemma 3.1. in [5]), it is easily seen that this indeed defines an equivalence relation on S. One can also verify that this equivalence relation is a strong equivalence relation. Note that, in this context, the group defined in (1) is exactly the commensurator of H in the group G. Moreover, the family F[H] defined in (2) consists of all finite subgroups of Comm_{G}[H] and all infinite virtually cyclic subgroups of Comm_{G}[H] that are equivalent to H. From Theorem 2.5, it now follows that in order to determine an upper bound for \underline{\underline{gd}}(G), we need find an upper bound for gd_{F[H]}(Comm_{G}[H]). This calls for a more careful study of the group Comm_{G}[H], which is the purpose of this section.

We begin by pointing out some basic properties of commensurators:

(i) If H and K are two equivalent infinite virtually cyclic subgroups of a group G, then their commensurators coincide. In particular, the commensurator of an infinite virtually cyclic subgroup H of G is the commensurator of any infinite cyclic subgroup of H.

(ii) If G is a subgroup of Γ such that H ⊆ G, then Comm_{G}[H] = Comm_{Γ}[H] ∩ G. Also, if π : G → Q is a group homomorphism such that |π(H)| = ∞, then π(Comm_{G}[H]) ⊆ Comm_{Q}[π(H)].

In the next results, we investigate the structure of commensurators of infinite virtually cyclic groups inside semi-direct products N ⋊ ℤ.

Lemma 3.1. Let N be a group and consider Γ = N ⋊ ℤ. Let H be an infinite virtually cyclic subgroup of Γ that is not contained in N. Then there is a semi-direct product Comm_{Γ}[H] = N_{0} ⋊_{φ} ⟨t⟩, with N_{0} a subgroup of N. Moreover, there exists an element a ∈ N_{0} and an integer n ∈ ℕ_{1} such that ⟨(a, t^{n})⟩ is an infinite cyclic subgroup of H and

(i) φ(a) = a;

(ii) for each x ∈ N_{0} there exists m ∈ ℕ_{1} such that φ^{nm}(x) = a^{−m}xa^{m}.

Proof. We may assume that H is infinite cyclic. Since H is contained in Comm_{Γ}[H] but not contained in N, we have a split short exact sequence,

1 → N_{0} → Comm_{Γ}[H] → ℤ = ⟨t⟩ → 0

determined by some φ ∈ Aut(N_{0}), where N_{0} is a subgroup of N. Let (b, t^{r}) be a generator of H, with r ∈ ℕ_{1}. Since (e, t) ∈ Comm_{Γ}[H], we must have that ⟨^{(e,t)}(b, t^{r})⟩ ∼ ⟨(b, t^{r})⟩. This implies that there exists a k ∈ ℕ_{1} such that

(e, t)(b, t^{r})^{k}(e, t^{−1}) = (b, t^{r})^{k}.

Denoting a = ∏_{i=0}^{k−1} φ^{ir}(b), the equation above implies that φ(a) = a. Let n = rk, then ⟨(a, t^{n})⟩ is an infinite cyclic subgroup of H such that φ(a) = a. Note that (a, t^{n})^{s} = (a^{s}, t^{ns}) for each s ∈ ℕ. Let x ∈ N_{0}. Since (x, 1) ∈ Comm_{Γ}[H] , we know there exists an m ∈ ℕ_{1} such that

(x, 1)(a, t^{n})^{m}(x^{−1}, 1) = (a, t^{n})^{m}.

This implies that (xa^{m}φ^{nm}(x)^{−1}, t^{nm}) = (a^{m}, t^{nm}), and hence φ^{nm}(x) = a^{−m}xa^{m}. □

Proposition 3.2. Let N be a group and consider Γ = N ⋊ ℤ. Let H be an infinite virtually cyclic subgroup of Γ that is not contained in N. Then the following hold.

(a) Every finitely generated subgroup of Comm_{Γ}[H] that contains H, is a semi-direct product of a finitely generated subgroup of N with ℤ.

(b) Let K be a finitely generated subgroup of Comm_{Γ}[H] that contains H. Then H has a finite index subgroup contained in the center of K.

Proof. Let K be a finitely generated subgroup of Comm_{Γ}[H] that contains H. By Lemma 3.1, we have that K = N_{0} ⋊_{φ} ⟨t⟩, where N_{0} a subgroup of N. Also by Lemma 3.1, there exists an element a ∈ N_{0} and an integer n ∈ ℕ_{1} such that ⟨(a, t^{n})⟩ is an infinite cyclic subgroup of H and for which φ(a) = a. Moreover, for each x ∈ N_{0} there exists m ∈ ℕ_{1} such that φ^{nm}(x) = a^{−m}xa^{m}. Since K is finitely generated, it can be generated by elements of the form

(x_{1}, 1), (x_{2}, 1), …, (x_{k}, 1), (e, t).

For each i ∈ {1, …, k}, take l_{i} ∈ ℕ_{1} such that φ^{nl_{i}}(x_{i}) = a^{−l_{i}}x_{i}a^{l_{i}}. Denote l = ∏_{i=1}^{k} l_{i}. Then φ^{ln}(x_{i}) = a^{−l}x_{i}a^{l} for all i ∈ {1, …, k}. Now, the set

{φ^{j}(x_{i}) | i ∈ {1, …, k}, j ∈ {0, …, ln − 1}} ∪ {a}

is a set of generators of N_{0} and ⟨(a^{l}, t^{nl})⟩ = ⟨(a, t^{n})^{l}⟩ is a finite index subgroup of H that is contained in the center of K. □

The previous proposition allows us to determine the structure of commensurators of infinite virtually cyclic groups inside certain elementary amenable groups.

Proposition 3.3. Let Λ be a locally finite group, N be a torsion-free nilpotent group, and

Λ → \overline{N} → N

be a short exact sequence of groups. Let Q be a virtually cyclic group and suppose that there is also a short exact sequence of groups

\overline{N} → G ⟶^{π} Q.

Let H be an infinite virtually cyclic subgroup of G such that Comm_{G}[H] = G and such that π(H) is a finite index subgroup of Q. Then every finitely generated subgroup of G is countable locally finite-by-virtually polycyclic.

Proof. We may assume that H is infinite cyclic. First consider the case where π(H), and hence Q, is finite. Let K be a finitely generated subgroup of G. Then P = K∩\overline{N} is a finite index subgroup of K and therefore also finitely generated. It follows that P is countable locally finite-by-poly-ℤ. Hence, K is countable locally finite-by-virtually polycyclic.

Now assume that π(H) is infinite. Let K be a finitely generated subgroup of G that contains H. Then P = K ∩ π^{−1}(π(H)) is a finite index subgroup of K that contains H and it is finitely generated. Since π(H) is infinite, we have π^{−1}(π(H)) ≅ \overline{N} ⋊ ℤ. This implies that P is isomorphic to a subgroup of a semi-direct product \overline{N} ⋊ ℤ such that H is not contained in \overline{N} and Comm_{\overline{N}⋊ℤ}[H] = \overline{N} ⋊ ℤ. It follows from Proposition 3.2(a) that P is countable locally finite-by-poly-ℤ. We conclude that K is countable locally finite-by-virtually polycyclic. This ends the proof, since every finitely generated subgroup of G is contained in a finitely generated subgroup of G that contains H. □

4. Elementary amenable groups

The class of elementary amenable groups is by definition the smallest class of groups that is closed under taking directed unions, extensions, subgroups and quotients, and that contains all finite groups and all abelian groups. This class of groups contains, for example, all virtually solvable groups. In [16], Hillman extended the notion of Hirsch length from polycyclic groups to elementary amenable groups. Its behaviour under taking subgroups, directed unions, and extensions is similar to that of for polycyclic groups. We refer the reader to section 1 in [16] for the precise definition of Hirsch length.

We shall only consider elementary amenable groups of finite Hirsch length. For these groups one has the following structure theorem.

Theorem 4.1 (Wehrfritz, [39, (g)], see also Hillman-Linnell, [17]). Let Γ be an elementary amenable group of finite Hirsch length h. There exist characteristic subgroups Λ(Γ) ⊆ N ⊆ M of Γ and a function j : ℕ → ℕ such that

(a) Λ(Γ) is the unique maximal normal locally finite subgroup of Γ, i.e. the locally finite radical;

(b) N/Λ(Γ) is torsion-free nilpotent; (c) M/N is free abelian of finite rank;

(d) [Γ : M] ≤ j(h).

Our task in this section is to prove Theorem A. But first, we need a few lemmas. We begin with a result that is closely related to Lemma 5.15 of [29].

Lemma 4.2. Let H be a infinite cyclic subgroup of a group Γ. If an infinite cyclic subgroup K of H is normal in Comm_{Γ}[H], then gd_{F[H]}(Comm_{Γ}[H]) = \underline{gd}(Comm_{Γ}[H]/K) and cd_{F[H]}(Comm_{Γ}[H]) = \underline{cd}(Comm_{Γ}[H]/K). If Comm_{Γ}[H] is virtually polycyclic, then H always has an infinite cyclic subgroup that is normal in Comm_{Γ}[H].

Proof. We may assume that H = K and that H is normal in Comm_{Γ}[H]. Consider the extension

(3)  0 → H → Comm_{Γ}[H] ⟶^{π} Comm_{Γ}[H]/H → 1.

Note that every group in F[H] is mapped onto a finite subgroup of Comm_{Γ}[H]/H by π. On the other hand, π^{−1}(F) ∈ F[H] for every finite subgroup F of Comm_{Γ}[H]/H. Therefore, it follows from Corollary 2.3 that gd_{F[H]}(Comm_{Γ}[H]) ≤ \underline{gd}(Comm_{Γ}[H]/H). On the other hand, if X is a model for E_{F[H]}Comm_{Γ}[H] then it is not difficult to verify that X^{H} will be a model for \underline{gd}(Comm_{Γ}[H]/H). We conclude that gd_{F[H]}(Comm_{Γ}[H]) = \underline{gd}(Comm_{Γ}[H]/H). To see that cd_{F[H]}(Comm_{Γ}[H]) = \underline{cd}(Comm_{Γ}[H]/H), let H be the family of finite subgroups of Comm_{Γ}[H]/H and consider the spectral sequence

E_{2}^{p,q}(M) = H_{H}^{p}(Comm_{Γ}[H]/H, H_{F[H]∩π^{−1}(−)}^{q}(π^{−1}(−), M)) ⟹ H_{F[H]}^{p+q}(Comm_{Γ}[H], M)

for M ∈ Mod-O_{F[H]}Comm_{Γ}[H], associated to extension (3) (see [30, Th. 5.1]). Since π^{−1}(F) ∈ F[H] for every finite subgroup F of Comm_{Γ}[H]/H, we have E_{2}^{p,q}(M) = 0 for all q > 1 and E_{2}^{n,0}(M) = H_{H}^{n}(Comm_{Γ}[H]/H, M(Comm_{Γ}[H]/π^{−1}(−))) . This implies that

H_{F[H]}^{n}(Comm_{Γ}[H], M) = H_{H}^{n}(Comm_{Γ}[H]/H, M(Comm_{Γ}[H]/π^{−1}(−)))

for every M ∈ Mod-O_{F[H]}Comm_{Γ}[H]. Let L ∈ Mod-O_{H}Comm_{Γ}[H]/H. Then we have L ∘ π ∈ Mod-_{O_{F} [H]}Comm_{Γ}[H] and

H_{F[H]}^{n}(Comm_{Γ}[H], L ∘ π) = H_{H}^{n}(Comm_{Γ}[H]/H, L)

for every L ∈ Mod-O_{H}Comm_{Γ}[H]/H. Therefore, we may conclude that

cd_{F[H]}(Comm_{Γ}[H]) = \underline{cd}(Comm_{Γ}[H]/H).

Now, let us assume that Comm_{Γ}[H] is virtually polycyclic. Let x_{1}, …, x_{r} be a set of generators for Comm_{Γ}[H] and let t be a generator for H. It follows that for every i ∈ {1, …, r}, we have m_{i}, n_{i} ∈ ℤ ∖ {0} such that x_{i}^{−1}t^{n_{i}}x_{i} = t^{m_{i}}. Lemma 5.14(v) in [29] implies that m_{i} = ±n_{i} for every i ∈ {1, …, r}. Define n = ∏_{i=1}^{r} n_{i}. One checks that x_{i}^{−1}t^{n}x_{i} = t^{±n} for all i ∈ {1, …, r}. This shows that ⟨t^{n}⟩ is a normal subgroup of Comm_{Γ}[H], and finishes the proof. □

Lemma 4.3. Let H be an infinite cyclic subgroup of a group Γ. If Comm_{Γ}[H] is countable locally finite-by-virtually cyclic, then Comm_{Γ}[H] is locally virtually cyclic and F[H] is the family of virtually cyclic subgroups of Comm_{Γ}[H]. In particular, we have that gd_{F[H]}(Comm_{Γ}[H]) ≤ 1.

Proof. By assumption, there is an extension

1 → L → Comm_{Γ}[H] ⟶^{π} V → 1,

where L is countable locally finite and V is virtually cyclic. Note that π maps H isomorphically onto a infinite cyclic subgroup S of V. Hence, π^{−1}(S) is a finite index subgroup of Comm_{Γ}[H] that is isomorphic to a semi-direct product L ⋊ H. It follows from Proposition 3.4.(b) of [5] that the family F[H] ∩ π^{−1}(S) coincides with the family of virtually cyclic subgroups of π^{−1}(S). Hence, F[H] is the family of virtually cyclic subgroups of Comm_{Γ}[H]. The remaining statements now follow from considering the finite index subgroup π^{−1}(S) of Comm_{Γ}[H], Proposition 3.4.(a) and Proposition 4.3. in [5] □

We are ready to prove Theorem A.

Proof of Theorem A. First note that the general case follows from the countable case and Corollary 2.4. Let Γ be a countable elementary amenable group of finite Hirsch length h. If h = 0 then Γ is locally finite, in which case \underline{\underline{gd}}(Γ) = \underline{gd}(Γ) ≤ 1 by Theorem 2.1. Now, assume that h ≥ 1. Let S be the set of infinite virtually cyclic subgroups of Γ and equip S with the equivalence relation discussed in Section 2. Let [H] be the equivalence class represented by H ∈ S and denote the set of equivalence classes by [S]. Let I be a complete set of representatives [H] of the Γ-orbits of [S] induced by conjugation. In [11], Flores and Nucinkis proved that \underline{cd}(Γ) ≤ h + 1. If h = 1, this implies that \underline{gd}(Γ) ≤ 3 = h + 2. If h ≥ 2 however, we can conclude that \underline{gd}(Γ) ≤ h + 1. Because of this discrepancy, we need to distinguish between two cases.

Case 1: h = 1. We claim that there exists a model for E_{F[H]}(Comm_{Γ}[H]) of dimension 1 and there is a model for \underline{E}(Comm_{Γ}[H]) of dimension 2, for each [H] ∈ I. To this end, consider [H] ∈ I. Let K be a finitely generated subgroup of Comm_{Γ}[H] that contains H. It follows from Theorem 4.1 that K is countable locally finite-by-virtually cyclic. Lemma 4.3 now implies that K is virtually cyclic. Hence, Comm_{Γ}[H] is locally virtually cyclic and the family F[H] coincides with the family of virtually cyclic subgroups of Comm_{Γ}[H]. Applying Corollary 2.4, our claims follow. Theorem 2.5 now implies that \underline{\underline{gd}}(Γ) ≤ 3 = h + 2.

Case 2: h ≥ 2. We claim that in this case, there exists a model for E_{F[H]}(Comm_{Γ}[H]) of dimension h + 1, for each [H] ∈ I. To prove this, consider [H] ∈ I and denote G = Comm_{Γ}[H]. It follows from Theorem 4.1 that there exists a countable locally finite group Λ, a torsion-free nilpotent group N and a finitely generated virtually abelian group A that fit into extensions

1 → Λ → \overline{N} → N → 1 and 1 → \overline{N} → G ⟶^{π} A → 1.

We will now consider two separate cases.

Case 2.a: π(H) is a finite subgroup of A. In this case, H has an infinite cyclic subgroup that is contained in \overline{N}, hence we may as well assume that H is infinite cyclic and contained in \overline{N}. Note that every group in F[H] is mapped onto a finite subgroup of A by π. Using Corollary 2.3, we obtain

gd_{F[H]}(G) ≤ \underline{gd}(A) + sup_{F ⊆A,|F |<∞}\{gd_{F[H]∩π^{−1}(F)}(π^{−1}(F))\}.

Let F be a finite subgroup of A and denote G_{0} = π^{−1}(F). Note that G_{0} contains H, that G_{0} = Comm_{G_{0}}[H] and that F[H] ∩ G_{0} consists of the finite subgroups of G_{0} and the infinite virtually cyclic subgroups of G_{0} that are equivalent to H. Let K be a finitely generated subgroup of G_{0} that contains H. It follows from Proposition 3.3 that there is a short exact sequence

1 → D → K ⟶^{p} Q → 1,

with D countable locally finite and Q virtually polycyclic of Hirsch length at most h(N). Note that p(H) is an infinite cyclic subgroup of Q. Let H be the family of subgroups of Q that contains all finite subgroup of Q and all infinite virtually cyclic subgroups of Q that are equivalent to p(H). Corollary 2.3 implies that

gd_{F[H]∩K}(K) ≤ gd_{H}Q + sup_{S∈H}{gd_{F[H]∩p^{−1}(S)}(p^{−1}(S))}.

Furthermore, it follows from Lemma 4.2 that

gd_{H}Q ≤ \underline{gd}(Q/p(H)) ≤ h(N) − 1

and by Lemma 4.3 that

sup_{S∈H}{gd_{F[H]∩p^{−1}(S)}(p^{−1}(S))} ≤ 1.

We can therefore conclude that gd_{F[H]∩K}(K) ≤ h(N). This implies that gd_{F[H]∩G_{0}}(G_{0}) ≤ h(N)+1, since every finitely generated subgroup of G_{0} is contained in a finitely generated subgroup of G_{0} that contains H. It follows that

max_{F ⊆A,|F |<∞}{gd_{F[H]∩π^{−1}(F)}(π^{−1}(F))} ≤ h(N) + 1.

Since \underline{gd}(A) = h(A), we find that gd_{F[H]}(G) ≤ h + 1.

Case 2.b: π(H) is an infinite subgroup of A. We may assume that H is infinite cyclic. Denote by H the family of subgroups of A that contains all finite subgroups of A and all infinite virtually cyclic subgroups that are equivalent to π(H). By Corollary 2.3, we have

gd_{F[H]}(G) ≤ gd_{H}(A) + max_{V ∈H}{gd_{F[H]∩π^{−1}(V)}(π^{−1}(V))}.

Since Comm_{A}[π(H)] = A and A is virtually polycyclic, by Lemma 4.2, we have

gd_{H}(A) = \underline{gd}(A/π(H)) ≤ h(A) − 1.

Let V ∈ H and denote π^{−1}(V) by G_{0}. Let K be a finitely generated subgroup of G_{0} that contains H. Again, by Proposition 3.3, we have a short exact sequence

1 → D → K ⟶^{p} Q → 1,

with D countable locally finite and Q virtually polycyclic. However, this time the Hirsch length of Q is at most h(N) + 1. Using the same approach as before, it follows that gd_{F[H]∩K}(K) ≤ h(N) + 1, and hence gd_{F[H]∩G_{0}}(G_{0}) ≤ h(N) + 2. We conclude that

max_{V ∈H}{gd_{F[H]∩π^{−1}(V)}(π^{−1}(V))} ≤ h(N) + 2

and gd_{F[H]}(G) ≤ h + 1. This proves the claim.

Since \underline{gd}(Γ) ≤ h + 1, it follows from Theorem 2.5 that \underline{\underline{gd}}(Γ) ≤ h + 2. □

Theorem A has the following corollary.

Corollary 4.4. Let Γ be a countable elementary amenable group of finite Hirsch length h, then

\underline{gd}(Γ) − 1 ≤ \underline{\underline{gd}}(Γ) ≤ \underline{gd}(Γ) + 1.

Proof. In the proof of Theorem A, we showed that gd_{F[H]}(Comm_{Γ}[H]) ≤ h + 1 for any infinite virtually cyclic subgroup H of Γ. In [11], Flores and Nucinkis established the inequality h ≤ \underline{cd}(Γ). Since this implies h ≤ \underline{gd}(Γ), it follows from Theorem 2.5 that \underline{\underline{gd}}(Γ) ≤ \underline{gd}(Γ) + 1. □

5. Examples

In this section, we will compute \underline{\underline{gd}}(Γ) for some elementary amenable groups Γ of finite Hirsch length. Our examples will illustrate that the bounds of Corollary 4.4 cannot be improved and all intermediate values can be attained. They will also show that the bounds established in Theorem A are attained for every h ≥ 1 and n ≥ 0.

First, we need a proposition.

Proposition 5.1. Let G be an infinite countable locally finite group and consider a semidirect product Γ = G ⋊_{φ} ℤ. Then cd_{ℚ}(Γ) = \underline{gd}(Γ) = 2.

Proof. Corollary 2.3 implies that \underline{gd}(Γ) ≤ 2. On the other hand, we have cd_{ℚ}(Γ) ≤ \underline{cd}(Γ) ≤ \underline{gd}(Γ). Therefore, it suffices to show that cd_{ℚ}(Γ) ≥ 2.

Suppose, by a way of contradiction, that cd_{ℚ}(Γ) ≤ 1. By Theorem 1.1 of [8], Γ acts on a tree T without edge inversions and with finite vertex stabilizers. Moreover, Corollary 1.2 of [8] implies that Γ is locally virtually cyclic. Now, fix a generator t of ℤ such that Γ = G ⋊_{φ} ⟨t⟩. Using the fact that Γ is locally virtually cyclic, one can check that for each g ∈ G there exists an l ∈ ℕ_{1} such that φ^{l}(g) = g. Hence, for each g ∈ G there exists an l ∈ ℕ_{1} such that t^{l} commutes with g in Γ. Since ⟨t⟩ acts freely on the tree T, it follows from Proposition 24 of [35], that T contains an infinite straight line L on which t acts by translation with fixed amplitude. Moreover, if for some k ∈ ℕ_{1}, a subtree of T is invariant under the action of ⟨t^{k}⟩, then it must contain L. Now, let g ∈ G and choose l ∈ ℕ_{1} such that t^{l} commutes with g. Since t^{l} and g commute, the line gL is invariant under the action of ⟨t^{l}⟩. It follows that L is contained in gL, so we must have gL = L. This shows that Γ acts on L. Since all vertex stabilizers are finite, Γ has a finite normal subgroup N such that Γ/N is isomorphic to a discrete subgroup of the isometries of the real line. Because every discrete subgroup of the isometry group of the real line is isomorphic to a subgroup of the infinite dihedral group, we conclude that Γ is virtually cyclic. This contradicts the fact that G is infinite locally finite. Therefore, we must have cd_{ℚ}(Γ) ≥ 2. □

In the next three examples we will construct elementary amenable groups G ⋊_{φ_{k}} ℤ, for k ∈ {−1, 0, 1} such that G is an infinite countable locally finite group and

\underline{\underline{gd}}(G ⋊_{φ_{k}} ℤ) = \underline{gd}(G ⋊_{φ_{k}} ℤ) + k.

In our first example, every element g ∈ G has a bounded orbit under the automorphism φ_{−1} ∈ Aut(G).

Example 5.2. Let G be an infinite countable locally finite group and let Γ be a semidirect product G⋊_{φ_{−1}} ℤ that is locally bounded, meaning that for each g ∈ G, there exists some l ∈ ℕ_{1} such that φ_{−1}^{l}(g) = g ( e.g. φ_{−1} = Id). Since Γ is not virtually cyclic, we have 1 ≤ \underline{\underline{gd}}(Γ). On the other hand, by [5, 4.3], we have \underline{\underline{gd}}(Γ) ≤ 1. We conclude that \underline{\underline{gd}}(Γ) = 1.

In the second example, due to M. Fluch, φ_{0} ∈ Aut(G) acts freely on the conjugacy classes of elements of G.

Example 5.3. Let F be a non-trivial finite group and consider the infinite direct sum G = ⨁_{i∈ℤ} F. One can form a semi-direct product G ⋊_{φ_{0}} ⟨t⟩ by letting t act on G via the shifting automorphism (i.e. t^{l} · (a_{i})_{i} = (a_{i−l})_{i} for all l ∈ ℤ). Then G ⋊_{φ_{0}} ⟨t⟩ is called the restricted wreath product of F and ℤ, which is denoted by F ≀ ℤ. In Example 18 in [12], Fluch shows that \underline{\underline{gd}}(F ≀ ℤ) = 2.

In our third example, the automorphism φ_{1} under consideration is a combination of the automorphisms of the two previous examples. This example also shows that the upper bound obtained in Theorem A of [5] is attained for h = 1 and any n ≥ 0.

Example 5.4. Let n be a non-negative integer, F be a non-trivial finite group, K be a non-trivial finite abelian group. Take the infinite direct sums A = ⨁_{i∈ℤ} F and B = ⨁_{ℵ_{n}} K. As before, consider the restricted wreath product F ≀ ℤ = A ⋊ ℤ. Let Γ = (F ≀ ℤ) ⊕ B. Clearly, Γ is an elementary amenable group of Hirsch length h = 1 and cardinality ℵ_{n}. Next, we will prove that \underline{\underline{gd}}(Γ) = n + 3.

Proof. First, we note that cd_{ℚ}(B) = n + 1 (see [7, 6.10]). Let M be a ℚ[B]-module such that H^{n+1}(B, M) ≠ 0. We can turn M into a Γ-module, via projection of Γ onto B. Let C be an infinite cyclic subgroup of F ≀ ℤ. We claim that the map

H^{n+2}(Γ, M) → H^{n+2}(Comm_{Γ}[C], M),

induced by the inclusion Comm_{Γ}[C] ⊆ Γ, is injective.

First of all, note that we may assume that C is maximal infinite cyclic. One can easily check that Comm_{Γ}[C] = Comm_{F ≀ℤ}[C] ⊕ B. It follows from Proposition 3.4.(a) of [5], that Comm_{F ≀ℤ}[C] is infinite cyclic. Since Comm_{F ≀ℤ}[C] contains C and C is maximal infinite cyclic, we have Comm_{F ≀ℤ}[C] = C. We conclude that Comm_{Γ}[C] = C ⊕ B. It now follows that Γ has a finite index subgroup isomorphic to (A ⋊ C) ⊕ B. A standard transfer-restriction argument now implies that the map

H^{n+2}(Γ, M) → H^{n+2}((A ⋊ C) ⊕ B, M),

induced by the inclusion (A ⋊ C) ⊕ B ⊆ Γ, is injective. Therefore, to prove the claim it suffices to show that the map

H^{n+2}((A ⋊ C) ⊕ B, M) → H^{n+2}(Comm_{Γ}[C], M),

induced by the inclusion Comm_{Γ}[C] = C ⊕ B ⊆ (A ⋊ C) ⊕ B is injective.

Let T be a ℚ[A ⋊ C]-module with trivial A-action. Since T is torsion-free and A is a torsion group that acts trivially on T, we have H^{1}(A, T) = 0. Using this together with the fact that cd_{ℚ}(C) and cd_{ℚ}(A) are both 1, an application of the Lyndon-Hochschild-Serre spectral sequence associated to the extension

1 → A → A ⋊ C → C → 1,

shows that H^{2}(A ⋊ C, T) = 0 for every ℚ[A ⋊ C]-module T with trivial A-action. These observations, together with the fact that cd_{ℚ}(B) = n + 1 and the naturality between the Lyndon-Hochschild-Serre spectral sequences with coefficients in M associated to the extensions

1 → B → (A ⋊ C) ⊕ B → A ⋊ C → 1

and

1 → B → Comm_{Γ}[C] → C → 1,

yield a commutative diagram

\xymatrix{H^{n+2}((A ⋊ C) ⊕ B, M) \ar[r] \ar[d]^{≅} & H^{n+2}(Comm_{Γ}[C], M) \ar[d]^{≅} \\ H^{1}(A ⋊ C, H^{n+1}(B, M)) \ar[r] & H^{1}(C, H^{n+1}(B, M)),}

where the horizontal maps are induced by the inclusions C ⊂ A⋊C and Comm_{Γ}[C] ⊆ (A⋊ C) ⊕ B and the vertical maps are isomorphisms arising form the spectral sequences. The Lyndon-Hochschild-Serre spectral sequence with coefficients in H^{n+1}(B, M) associated to the extension

1 → A → A ⋊ C → C → 1

implies that we have a short exact sequence

0 → H^{1}(C, H^{n+1}(B, M)) → H^{1}(A ⋊ C, H^{n+1}(B, M)) → H^{0}(C, H^{1}(A, H^{n+1}(B, M))) → 0.

Since H^{n+1}(B, M) is torsion-free and A is a torsion group that acts trivially on H^{n+1}(B, M), we have H^{1}(A, H^{n+1}(B, M)) = 0 and therefore

H^{1}(A ⋊ C, H^{n+1}(B, M)) ≅ H^{1}(C, H^{n+1}(B, M)).

Using this, we obtain a commutative diagram

\xymatrix{H^{n+2}((A ⋊ C) ⊕ B, M) \ar[r] \ar[d]^{≅} & H^{n+2}(Comm_{Γ}[C], M) \ar[d]^{≅} \\ H^{1}(C, H^{n+1}(B, M)) \ar[r]^{Id} & H^{1}(C, H^{n+1}(B, M)).}

that proves our claim.

Since H^{∗}(F, M) = 0 for all finite subgroups F of Γ, we may conclude from Theorem 6.1 in [30] that there are natural isomorphisms

H_{F}^{∗}(Γ, M^{−}) ≅ H^{∗}(Γ, M)

and

H_{F∩Comm_{Γ}[C]}^{∗}(Comm_{Γ}[C], M^{−}) ≅ H^{∗}(Comm_{Γ}[C], M).

Hence, for every infinite cyclic subgroup C of F ≀ ℤ, the map

i^{n+2}(C) : H_{F}^{n+2}(Γ, M^{−}) → H_{F∩Comm_{Γ}[C]}^{n+2}(Comm_{Γ}[C], M^{−})

is injective and

H_{F∩Comm_{Γ}[C]}^{n+2}(Comm_{Γ}[C], M^{−}) ≅ H^{1}(C, H^{n+1}(B, M)) ≅ H^{n+1}(B, M) ≠ 0.

Next, observe that cd_{ℚ}(ℤ ⊕ B) = n + 2 and since ℤ ⊕ B is a subgroup of Γ, it follows that cd_{ℚ}(Γ) ≥ n + 2. But, by Corollary 2.4, we have that \underline{gd}(Γ) ≤ n + 2. Therefore, \underline{gd}(Γ) = n + 2.

We also note that from Corollary 2.4(ii) and the proof of Theorem A (Case 1: h = 1) that gd_{F[H]}(Comm_{Γ}[H]) ≤ n + 1 for all infinite virtually cyclic subgroups H of Γ.

Let H be the family of virtually cyclic subgroups of Γ. Proposition 7.1 gives us an exact sequence

H_{F}^{n+2}(Γ, M^{−}) \underrightarrow{∏ i^{n+2}(H)} ∏_{[H]∈I} H_{F∩Comm_{Γ}[H]}^{n+2}(Comm_{Γ}[H], M^{−}) → H_{H}^{n+3}(Γ, M^{−}) → 0.

Since the set I contains more than one element represented by an infinite cyclic subgroup of F ≀ ℤ, the map ∏ i^{n+2}(H) cannot be surjective because i^{n+2}(H) is injective when H is a subgroup F ≀ ℤ. Since

∏_{[H]∈I} H_{F∩Comm_{Γ}[H]}^{n+2}(Comm_{Γ}[H], M^{−}) ≠ 0,

it follows that H_{H}^{n+3}(Γ, M^{−}) ≠ 0. Therefore, we have \underline{\underline{gd}}(Γ) ≥ n + 3. Theorem A now implies that \underline{\underline{gd}}(Γ) = n + 3. □

It is well-known and not difficult to see that, for any two groups Γ_{1} and Γ_{2}, one has

\underline{gd}(Γ_{1} × Γ_{2}) ≤ \underline{gd}(Γ_{1}) + \underline{gd}(Γ_{2}).

When considering the family of virtually cyclic subgroups, the bounds are not as clear when taking products. As pointed out in [29], Corollary 5.6 and Remark 5.7, the best one can do in general is

\underline{\underline{gd}}(Γ_{1} × Γ_{2}) ≤ \underline{\underline{gd}}(Γ_{1}) + \underline{\underline{gd}}(Γ_{2}) + 3.

Next, we will consider two examples that will again illustrate the somewhat strange behaviour of this geometric dimension when taking products of groups.

Example 5.5. Let F be a non-trivial finite abelian group. Then for any integers n ≥ 0 and h ≥ 2, we have \underline{\underline{gd}}((⨁_{ℵ_{n}} F) ⊕ ℤ^{h}) = n + h + 2 and \underline{\underline{gd}}((⨁_{ℵ_{n}} F) ⊕ ℤ) = n + 1.

Proof. Let Γ = (⨁_{ℵ_{n}} F) ⊕ ℤ^{h}. Then Γ is elementary amenable of Hirsch length h and cardinality ℵ_{n}. We will prove that \underline{\underline{gd}}(Γ) = n + h + 2. By Theorem A, it suffices to show that \underline{\underline{cd}}(Γ) ≥ n + h + 2.

By Corollary 6.10 of [7], cd_{ℚ}(⨁_{ℵ_{n}} F) = n + 1. So, a corner argument of the Lyndon-Hochschild-Serre spectral sequence associated to the extension with ⨁_{ℵ_{n}} F as its kernel, implies cd_{ℚ}(Γ) = n+h+1. Since \underline{gd}(Γ) ≤ n+h+1 (see Corollary 2.4(ii) and [11, Corollary 4]), this implies that \underline{gd}(Γ) = n + h + 1. Note that for every infinite cyclic subgroup H of Γ, Comm_{Γ}[H] = Γ. Also, by Lemma 4.2, we have that gd_{F[H]}(Comm_{Γ}[H]) ≤ n + h.

Next, let H be the family of virtually cyclic subgroups of Γ and let F be the family of finite subgroups of Γ. Since \underline{gd}(Γ) = n + h + 1, we can find a module M ∈ Mod-O_{F}Γ such that H_{F}^{n+h+1}(Γ, M) ≠ 0. Denote the induction functor from Mod-O_{F}Γ to Mod-O_{H}Γ by ind_{F}^{H} and note that by Lemma 7.3 ind_{F}^{H}(M) ≅ M as O_{F}Γ-modules. Proposition 7.1 gives an exact sequence

( ∏_{[H]∈I} H_{F[H]}^{n+h+1}(Comm_{Γ}[H], ind_{F}^{H}(M)))⊕H_{F}^{n+h+1}(Γ, M) → ∏_{[H]∈I} H_{F∩Comm_{Γ}[H]}^{n+h+1}(Comm_{Γ}[H], M)

→ H_{H}^{n+h+2}(Γ, ind_{F}^{H}(M)) → ( ∏_{[H]∈I} H_{F[H]}^{n+h+2}(Comm_{Γ}[H], ind_{F}^{H}(M))) ⊕ H_{F}^{n+h+2}(Γ, M)

which, by the above established facts, transforms to

H_{F}^{n+h+1}(Γ, M) ⟶^{Δ} ∏_{[H]∈I} H_{F}^{n+h+1}(Γ, M) → H_{H}^{n+h+2}(Γ, ind_{F}^{H}(M)) → 0.

Since H_{F}^{n+h+1}(Γ, M) is non-zero, Δ is the diagonal map and I contains more than one element (because h ≥ 2), it follows that Δ cannot be surjective. This implies that H_{H}^{n+h+2}(Γ, ind_{F}^{H}(M)) ≠ 0 and hence \underline{\underline{cd}}(Γ) ≥ n+h+2. We conclude that \underline{\underline{gd}}(Γ) = n+h+2.

Let G = (⨁_{ℵ_{n}} F) ⊕ ℤ. Note that all infinite cyclic subgroups of G are equivalent and G = Comm_{G}[ℤ]. Using Lemma 4.2, it follows that

\underline{\underline{gd}}(G) = gd_{F[ℤ]}(Comm_{G}[ℤ]) = \underline{gd}(⨁_{ℵ_{n}} F).

Since cd_{ℚ}(⨁_{ℵ_{n}} F) = n + 1, applying Theorem 2.1 yields \underline{gd}(⨁_{ℵ_{n}} F) = n + 1. Hence, \underline{\underline{gd}}(G) = n + 1. □

Example 5.6. Let F be a non-trivial finite group. Then \underline{\underline{gd}}((F ≀ ℤ) ⊕ ℤ) = 3 while \underline{\underline{gd}}(F ≀ ℤ) = 2 and \underline{\underline{gd}}(ℤ) = 0.

Proof. Let Γ = (F ≀ ℤ) ⊕ ℤ. Since cd_{ℚ}(F ≀ ℤ) = 2, one can deduce that \underline{gd}(Γ) = 3 by showing that cd_{ℚ}(Γ) = 3 using a spectral sequence argument. We claim that \underline{\underline{gd}}(Γ) = 3. Note that ℤ^{2} is a subgroup of Γ, therefore \underline{\underline{gd}}(Γ) ≥ 3. Hence, to prove the claim it suffices to show that \underline{\underline{cd}}(Γ) ≤ 3. Let H be the family of virtually cyclic subgroups of Γ and let F be the family of finite subgroups of Γ. Let M ∈ Mod-O_{H}Γ. By Proposition 7.1, we have an exact sequence

(4) ( ∏_{[H]∈I} H_{F[H]}^{3}(Comm_{Γ}[H], M)) ⊕ H_{F}^{3}(Γ, M) → ∏_{[H]∈I} H_{F∩Comm_{Γ}[H]}^{3}(Comm_{Γ}[H], M)

→ H_{H}^{4}(Γ, M) → ( ∏_{[H]∈I} H_{F[H]}^{4}(Comm_{Γ}[H], M)) ⊕ H_{F}^{4}(Γ, M).

Let H be an infinite cyclic subgroup of Γ. If H is contained in the direct summand ℤ of Γ, then clearly Comm_{Γ}[H] = Γ. Using Corollary 5.2 in [30] one checks that cd_{F[H]}(Comm_{Γ}[H]) ≤ \underline{cd}(F ≀ ℤ). Hence, we have cd_{F[H]}(Comm_{Γ}[H]) ≤ \underline{gd}(F ≀ ℤ) = 2. Furthermore, there is exactly one [H] ∈ I such that H is an infinite cyclic subgroup of Γ that is contained in the direct summand ℤ of Γ.

Now suppose H is an infinite cyclic subgroup of Γ that is not equivalent to the direct summand ℤ of Γ. One can verify, via direct computation or by using Proposition 3.4(a) of [5], that Comm_{Γ}[H]∩(F ≀ℤ) ≅ ℤ. This implies that Comm_{Γ}[H] ≅ ℤ^{2}, and hence \underline{cd}(Comm_{Γ}[H]) = 2. Let C be the maximal infinite cyclic subgroup of Comm_{Γ}[H] containing H. The subgroups in F[H] are exactly the subgroups of C, and Comm_{Γ}[H]/C ≅ ℤ.

Hence, we have

cd_{F[H]}(Comm_{Γ}[H]) ≤ cd(Comm_{Γ}[H]/C) = cd(ℤ) = 1

by applying Corollary 5.2 in [30] to the extension 0 → C → Comm_{Γ}[H] → ℤ → 0, with the appropriate families. Since \underline{gd}(Γ) = 3, the exact sequence (4) becomes

H_{F}^{3}(Γ, M) ⟶^{Id} H_{F}^{3}(Γ, M) → H_{H}^{4}(Γ, M) → 0.

This implies that H_{H}^{4}(Γ, M) = 0, so \underline{\underline{cd}}(Γ) ≤ 3. We conclude that \underline{\underline{gd}}(Γ) = 3. □

6. Extensions with torsion-free quotient

The following corollary follows easily from Theorem A and Corollary 2.3.

Corollary 6.1. Consider a short exact sequence of groups N → Γ → Q such that N is elementary amenable of finite Hirsch length and of cardinality ℵ_{n} and such that \underline{\underline{gd}}(Q) < ∞. Then

\underline{\underline{gd}}(Γ) ≤ \underline{\underline{gd}}(Q) + n + h(N) + 3.

In general, given a short exact sequence of groups N → Γ → Q, one can ask whether or not the finiteness of \underline{\underline{gd}}(N) and \underline{\underline{gd}}(Q) implies the finiteness of \underline{\underline{gd}}(Γ). The answer to this question is unknown in general. Theorem B gives a positive answer when Q is torsion-free and the number δ(N) is finite.

We start with two lemmas.

Lemma 6.2. Let N be a group of cardinality ℵ_{n}, then \underline{gd}(N) ≤ δ(N) + 1.

Proof. This is immediate from the definition of δ(N) and Corollary 2.4(i). □

Lemma 6.3. Let H be an infinite virtually cyclic subgroup of a group Γ, then

cd_{F[H]}(Comm_{Γ}[H]) ≤ max\{\underline{\underline{cd}}(Comm_{Γ}[H]), \underline{cd}(Comm_{Γ}[H])\}.

Proof. Let G = Comm_{Γ}[H] and denote by H the family of virtually cyclic subgroups of G. Take K ∈ Mod-O_{F[H]}G and denote the induction functor from Mod-O_{F[H]}G to Mod-O_{H}G by ind_{F[H]}^{H}. Let M = ind_{F[H]}^{H}(K). By Proposition 7.1, we have an exact sequence

H_{H}^{n}(G, M) → ( ∏_{[S]∈I} H_{F[S]}^{n}(Comm_{G}[S], M))⊕H_{F}^{n}(G, M) → ∏_{[S]∈I} H_{F∩Comm_{G}[S]}^{n}(Comm_{G}[S], M).

Note that [H] ∈ I, and that Comm_{G}[H] = G. Hence, if we choose n > max\{\underline{\underline{cd}}(G), \underline{cd}(G)\}, it follows from the exact sequence and Lemma 7.3 that H_{F[H]}^{n}(G, M) = H_{F[H]}^{n}(G, K) = 0. Since this holds for any K ∈ Mod-O_{F[H]}G, we conclude that

cd_{F[H]}(Comm_{G}[H]) ≤ max\{\underline{\underline{cd}}(Comm_{Γ}[H]), \underline{cd}(Comm_{Γ}[H])\}.
□

We can now prove Theorem B.

Proof of Theorem B. It suffices to prove that for any semi-direct product N ⋊ ℤ, we have

\underline{\underline{gd}}(N ⋊ ℤ) ≤ max\{\underline{\underline{gd}}(N), δ(N)\} + 2.

Because then it would follow from Corollary 2.3 that

\underline{\underline{gd}}(Γ) ≤ \underline{\underline{gd}}(Q) + max\{\underline{\underline{gd}}(N), δ(N)\} + 2,

since Q is torsion-free.

If N is virtually cyclic, then the desired inequality clearly holds. Assume that N is not virtually cyclic. Then \underline{\underline{gd}}(N) ≥ 1, so it suffices to prove that

\underline{\underline{cd}}(N ⋊ ℤ) ≤ max\{\underline{\underline{gd}}(N), δ(N)\} + 2.

Let us consider a semi-direct product G = N ⋊ ℤ and an infinite cyclic subgroup H of G. There is a short exact sequence

1 → N_{0} → Comm_{G}[H] ⟶^{p} mℤ → 0

where N_{0} = N ∩ Comm_{G}[H] and m is an integer that can be zero. We will find an upper bound for cd_{F[H]}(Comm_{G}[H]). To do this, we distinguish between two cases.

First, assume that H is contained in N. If K is a virtually cyclic subgroup of Comm_{G}[H] which is equivalent to H then p(K) is a finite group, since H is contained in the kernel of p. Since mℤ is torsion-free, we conclude that every subgroup of F[H] is contained in N_{0}. Since N_{0} = Comm_{N_{0}}[H], it follows from Lemma 6.3 that

cd_{F[H]}(N_{0}) ≤ max\{\underline{\underline{cd}}(N_{0}), \underline{cd}(N_{0})\} ≤ max\{\underline{\underline{gd}}(N), \underline{gd}(N)\}.

Hence, Corollary 2.3 implies that

cd_{F[H]}(Comm_{G}[H]) ≤ max\{\underline{\underline{gd}}(N), \underline{gd}(N)\} + 1.

Now, assume that H is not contained in N (in this case m is not zero). Let K be a finitely generated subgroup of Comm_{G}[H] that contains H. By Proposition 3.2, K is a Bby-ℤ, where B is a finitely generated subgroup of N, and H has an infinite cyclic subgroup R that is central in K. It follows from Lemma 4.2 that gd_{F[H]∩K}(K) = \underline{gd}(K/R). Since K/R is B-by-finite cyclic where B is a finitely generated subgroup of N, it follows that gd_{F[H]∩K}(K) ≤ δ(N)−n. Hence, by Corollary 2.4 we have gd_{F[H]}(Comm_{G}[H]) ≤ δ(N)+1. Therefore, in general we have

cd_{F[H]}(Comm_{G}[H]) ≤ max\{\underline{\underline{gd}}(N), \underline{gd}(N), δ(N)\} + 1.

Using Theorem 7.2 together with Lemma 6.2, we deduce that

\underline{\underline{cd}}(N ⋊ ℤ) ≤ max\{\underline{\underline{gd}}(N), δ(N)\} + 2.
□

As mentioned in the introduction, we do not know of an example of a group N such that \underline{gd}(N) < ∞ and δ(N) = ∞. However, we will show that there are groups N for which δ(N) is finite but the difference δ(N) − \underline{gd}(N) is arbitrarily large. This groups will be constructed by the means of Theorem C.

Before going into the proof of this theorem, let us first recall some of the basics of the general construction of Bestvina-Brady groups H_{L} and the semi-direct products H_{L} ⋊ Q. For more details, we suggest the original sources [2] and [22].

For a given finite flag complex L, the right-angled Artin group G_{L} is the group whose generators are in one-to-one correspondence with the vertex set of L subject only to the relations that if two vertices are connected by an edge in the 1-skeleton of L, they commute. There is an epimorphism ϕ_{L}: G_{L} → ℤ defined by mapping each of the given generators to 1. The group H_{L} is defined to be the kernel of ϕ_{L}. It is a general fact that any right-angled Artin group can be embedded into a right angled Coxeter group which in turn is known to be a subgroup of GL_{m}(ℤ) for some m (see [22, Corollary 8]). This shows that H_{L} is an integral linear group. The map sending L to G_{L} is a functor from flag complexes to groups. Therefore, every automorphism of L induces an automorphism of G_{L}. In particular, if a group Q acts admissibly and simpicially on L, this induces a representation of Q into the group of automorphisms of G_{L}. Since the epimorphism ϕ_{L} is Q-equivariant, this representation restricts to H_{L} and one can form the semi-direct product H_{L} ⋊ Q.

Proof of Theorem C. (a). Let L be an acyclic n-dimensional finite flag complex with a simplicial admissible faithful action of a cyclic group Q of a prime order. It follows from the Main Theorem of [2] that H_{L} is of type F P and if one imposes the stronger condition that L is contractible, then H_{L} is of type F. Hence Γ = H_{L}⋊Q is of type V F P in general, and of type V F if L is contractible. Since L is acyclic, it follows from Theorem 22 of [23] that cd(H_{L}) = n. Since H_{L} contains ℤ^{n} as a subgroup, we also have hd(H_{L}) = n. Denote by C_{Γ}(Q) the centralizer of Q in Γ and set C_{H_{L}}(Q) = C_{Γ}(Q) ∩ H_{L}. Note that the normalizer N_{Γ}(Q) of Q in Γ is isomorphic to C_{H_{L}}(Q) × Q and that the Weyl group W (Q) = N_{Γ}(Q)/Q is isomorphic to C_{H_{L}}(Q). Since, by Smith Theory (see [36]), the fixedpoint set L^{Q} is non-empty, from Theorem 3(2) of [22], it follows that all non-trivial finite subgroups of Γ are conjugate to Q and that C_{H_{L}}(Q) is the Bestvina-Brady group H_{L^{Q}}.

Recall that in our case Γ_{s} = ∏_{i=1}^{s} H_{L_{i}} ⋊ C_{p_{i}} where H_{L_{i}} is the Bestvina-Brady group associated to the n_{i}-dimensional acyclic finite flag complex L_{i} for i = 1, …, s. So, in the above, replacing L and Q with L_{i} and C_{p_{i}}, respectively, for each i, we obtain that H_{L_{i}} is an F P -group, ℤ^{n_{i}} is a subgroup of H_{L_{i}}, and hd(H_{L_{i}}) = cd(H_{L_{i}}) = n_{i}. This implies that Γ_{s} is of type V F P (V F if L is contractible) and vhd(Γ_{s}) = vcd(Γ) = ∑_{i=1}^{s} n_{i}. Note that each group H_{L_{i}} ⋊ C_{p_{i}} is an integral linear group because it is a finite extension of such a group. Since Γ_{s} is a finite product of integral linear groups, it is itself integral linear.

For the rest of the proof we argue by induction on s. Suppose s = 1. Denote p = p_{1}, n_{1} = n, L = L_{1} and Γ = Γ_{1}. Let F be the family of finite subgroups of Γ, which in this case consists of the trivial subgroup and subgroups isomorphic to C_{p}. Let T be the family containing only the trivial subgroup of Γ. Define the covariant functor

V : O_{F}Γ → ℤ-mod : Γ/H ↦ \{ \begin{matrix}𝔽_{q} & if & H = \{e\} \\ 0 & if & H ≠ \{e\}, \end{matrix}

where every morphism Γ/H ⟶^{g} Γ/K with K ≠ {e} is mapped to the zero homomorphism and every morphism Γ/{e} ⟶^{g} Γ/{e} is mapped to the identity homomorphism. We define a strong equivalence relation on F ∖ T by

H ∼ K ⇔ H = K.

Under this equivalence relation, one has N_{Γ}[H] = C_{Γ}(H), the family T [H] is {{e}, H}, and there is exactly one conjugacy class of equivalence classes of non-trivial subgroups of Γ, namely the one represented by C_{p}. Now, the Tor long exact sequence of Theorem 7.7 with M = \underline{ℤ} and T = V yields the exact sequence

… → H_{n+1}^{F}(Γ, V) → H_{n}(C_{Γ}(C_{p}), 𝔽_{q}) → H_{n}^{T [C_{p}]}(C_{Γ}(C_{p}), V) ⊕ H_{n}(Γ, 𝔽_{q}) → ….

Since C_{Γ}(C_{p}) ≅ C_{H_{L}}(C_{p}) × C_{p} and p ≠ q, it follows that

H_{n}(C_{Γ}(C_{p}), 𝔽_{q}) = H_{n}(C_{H_{L}}(C_{p}), 𝔽_{q}).

We claim that H_{n}^{T [C_{p}]}(C_{Γ}(C_{p}), V) = 0. To see this, consider the convergent E_{2}-term spectral sequence

H_{t}(C_{H_{L}}(C_{p}), H_{r}^{T [C_{p}]}(C_{p}, V)) ⟹ H_{t+r}^{T [C_{p}]}(C_{Γ}(C_{p}), V)

associated to the extension

1 → C_{p} → C_{Γ}(C_{p}) → C_{H_{L}}(H) → 1

(see [30, Theorem 5.1]). Since C_{p} ∈ T [C_{p}], we have H_{r}^{T [C_{p}]}(C_{p}, V) = 0 for all r ≥ 1 and H_{0}^{T [C_{p}]}(C_{p}, V) = V (Γ/C_{p}) = 0. Hence, all the E_{2}-terms vanish which proves the claim. We conclude that there is an exact sequence

H_{n+1}^{F}(Γ, V) → H_{n}(C_{H_{L}}(C_{p}), 𝔽_{q}) → H_{n}(Γ, 𝔽_{q}).

Since Γ is of type F P_{∞}, the group H_{n}(Γ, 𝔽_{q}) is a finite dimensional vector space over 𝔽_{q}. However, combining Corollary 7 in [23] together with the hypothesis that H_{n−1}(L^{C_{p}}, 𝔽_{q}) ≠ 0, shows that H_{n}(C_{H_{L}}(C_{p}), 𝔽_{q}) is an infinite dimensional vector space over 𝔽_{q}. In follows that H_{n+1}^{F}(Γ, V) is an infinite dimensional vector space over 𝔽_{q} and in particular, \underline{hd}(Γ) ≥ n+1. Since \underline{cd}(Γ) ≤ n+1, by Theorem 6.4 of [25], it follows that \underline{hd}(Γ)) = \underline{cd}(Γ) = n+1. This concludes the case s = 1.

For each 2 ≤ k ≤ s, let F_{k} be the family of finite subgroups of Γ_{k} = ∏_{i=1}^{k} H_{L_{i}} ⋊ C_{p_{i}}. By Theorem 6.4 in [25], we have that

\underline{cd}(Γ_{k}) ≤ ∑_{i=1}^{k}(n_{i} + 1) = k + ∑_{i=1}^{k} n_{i}.

We will show that \underline{hd}(Γ_{k}) ≥ k + ∑_{i=1}^{k} n_{i}, which will imply that

\underline{hd}(Γ_{k}) = \underline{cd}(Γ_{k}) = k + ∑_{i=1}^{k} n_{i}.

To this end, define the covariant functor

V_{k} : O_{F_{k}}Γ_{k} → ℤ-mod : Γ_{k}/H ↦ \{ \begin{matrix}𝔽_{q} & if & H = \{e\} \\ 0 & if & H ≠ \{e\}, \end{matrix}

where every morphism Γ_{k}/H ⟶^{g} Γ_{k}/K with K ≠ {e} is mapped to the zero homomorphism, and every morphism Γ_{k}/{e} ⟶^{g} Γ_{k}/{e} is mapped to the identity homomorphism. To finish the proof, it suffices to show that H_{m_{k}}^{F_{k}}(Γ_{k}, V_{k}) is an infinite dimensional 𝔽_{q}-vector space where m_{k} = k + ∑_{i=1}^{k} n_{i}.

We denote Γ = H_{L_{k}} ⋊ C_{p_{k}}, define F to be the family of finite subgroups of Γ, and let V : O_{F}G → ℤ-mod be the covariant functor as before. Note that G_{k} = Γ_{k−1} × Γ. Associated to the extension

1 → Γ → Γ_{k} ⟶^{π} Γ_{k−1} → 1

there is a convergent E_{2}-term spectral sequence (see [30, 5.1])

(5)  H_{s}^{F_{k−1}}(Γ_{k−1}, H_{r}^{F_{k}∩π^{−1}(−)}(π^{−1}(−), V_{k})) ⟹ H_{s+r}^{F_{k}}(Γ_{k}, V_{k}).

Observe that for any K ∈ F_{k−1}, the preimage π^{−1}(K) is direct product of Γ with a finite abelian group A. Applying the spectral sequence of [30, 5.1] associated to the extension 1 → A → π^{−1}(K) ⟶^{ω} Γ → 1 gives us

H_{r}^{F_{k}∩π^{−1}(K)}(π^{−1}(K), V_{k}) ≅ H_{r}^{F_{k}∩Γ}(Γ, H_{0}^{F_{k}∩ω^{−1}(−)}(ω^{−1}(−), V_{k}))
≅ H_{r}^{F_{k}∩Γ}(Γ, V_{k}(Γ_{k}/ω^{−1}(−)))
≅ H_{r}^{F}(Γ, V_{k}(Γ_{k}/ω^{−1}(−)))

where first and second isomorphisms follow from the fact that ω^{−1}(F) ∈ F_{k} for every finite subgroup F of Γ. One can now easily check that the functor H_{r}^{F_{k}∩π^{−1}(−)}(π^{−1}(−), V_{k}) is isomorphic to the functor V_{k−1} ⊗_{ℤ} H_{r}^{F}(Γ, V).

Next, using a corner argument for the spectral sequence (5), we obtain

H_{m_{k}}^{F_{k}}(Γ_{k}, V_{k}) = H_{m_{k−1}}^{F_{k−1}}(Γ_{k−1}, H_{n_{k}+1}^{F_{k}∩π^{−1}(−)}(π^{−1}(−), V_{k}))
≅ H_{m_{k−1}}^{F_{k−1}}(Γ_{k−1}, V_{k−1} ⊗_{ℤ} H_{n_{k}+1}^{F}(Γ, V)).

Now, let P_{∗} → \underline{ℤ} be a free contravariant O_{F_{k−1}}Γ_{k−1}-resolution of \underline{ℤ}. Then C_{∗} = P_{∗} ⊗_{F_{k−1}} V_{k−1} is a complex of 𝔽_{q}-vector spaces. One verifies that

P_{∗} ⊗_{F_{k−1}} (V_{k−1} ⊗_{ℤ} H_{n_{k}+1}^{F}(Γ, V)) ≅ C_{∗} ⊗_{ℤ} H_{n_{k}+1}^{F}(Γ, V).

This implies that

H_{m_{k−1}}^{F_{k−1}}(Γ_{k−1}, V_{k−1} ⊗_{ℤ} H_{n_{k}+1}^{F}(Γ, V)) = H_{m_{k−1}}(C_{∗} ⊗_{ℤ} H_{n_{k}+1}^{F}(Γ, V))
= H_{m_{k−1}}(C_{∗}) ⊗_{ℤ} H_{n_{k}+1}^{F}(Γ, V)
= H_{m_{k−1}}^{F_{k−1}}(Γ_{k−1}, V_{k−1}) ⊗_{ℤ} H_{n_{k}+1}^{F}(Γ, V).

This second equality follow from the fact that C_{∗} and H_{n_{k}+1}^{F}(Γ, V) are both vector spaces over 𝔽_{q}, so tensoring with H_{n_{k}+1}^{F}(Γ, V) is exact. We conclude by induction that H_{m_{k}}^{F_{k}}(G_{k}, V_{k}) is an infinite dimensional vector space over 𝔽_{q}. This finished the proof of part (a).

(b). Since Λ_{s} is an extension of N = ∏_{i=1}^{s} H_{L_{i}} by ℤ it follow that

cd(Λ_{s}) ≤ 1 + cd(N) = 1 + ∑_{i=1}^{s} n_{i}.

On the other hand, since each Bestvina-Brady group H_{L_{i}} contains a subgroup isomorphic to ℤ^{n_{i}}, it is not difficult to see that Λ_{s} contains a free abelian subgroups of rank 1+∑_{i=1}^{s} n_{i}. This shows that 1 + ∑_{i=1}^{s} n_{i} ≤ hd(Λ_{s}). Since hd(Λ_{s}) ≤ cd(Λ_{s}), we deduce that

hd(Λ_{s}) = cd(Λ_{s}) = 1 + ∑_{i=1}^{s} n_{i}.

Recall that right-angled Artin groups are CAT(0)-groups and hence, by Theorem 1.1 of [27], have finite dimensional classifying spaces for the family of virtually cyclic subgroups. Since N is an s-fold product of subgroups of right-angled Artin groups, it follows for instance from Corollary 5.6 of [29] that \underline{\underline{gd}}(N ×ℤ) < ∞. Because Λ_{s} contains a finite index subgroup isomorphic to N × ℤ, we conclude from Theorem 2.4 of [25] that \underline{\underline{gd}}(Λ_{s}) < ∞. Hence, \underline{\underline{cd}}(Λ_{s}) < ∞.

By part (a), we have cd(Γ_{s}) = s + cd(N) with s ≥ 2. So, Lemma 6.4 below gives us

\underline{\underline{cd}}(Λ_{s}) ≥ s + cd(N) = s + ∑_{i=1}^{s} n_{i}.

Finally, since Λ_{s} is a torsion-free finite extension of N × ℤ which is an integral linear group of type F P, it follows that Λ_{s} is an integral linear group of type F P. If additionally L is contractible, then Λ_{s} is of type F. □

Lemma 6.4. Let N → Γ ⟶^{π} C_{d} be a short exact sequence of groups such that \underline{cd}(Γ) ≥ \underline{cd}(N) + s for some s ≥ 2. Let Λ = N ⋊ ℤ be the pull-back of Γ under projection of ℤ onto C_{d}. Then \underline{\underline{cd}}(Λ) ≥ \underline{cd}(N) + s.

Proof. The group Λ fits into the following commutative diagram with exact rows and surjective vertical maps

\xymatrix{1 \ar[r] & N \ar[r] \ar[d]^{Id} & Λ \ar[r] \ar[d] & ℤ \ar[d] \ar[r] & 1 \\ 1 \ar[r] & N \ar[r] & Γ \ar[r]^{π} & C_{d} \ar[r] & 1.}

Note that Λ has a infinite cyclic normal subgroup H such that Λ/H = Comm_{Λ}[H]/H = Γ. Therefore, it follows from Lemma 4.2 that

cd_{F[H]}(Λ) = cd_{F[H]}(Comm_{Λ}[H]) = \underline{cd}(Γ) ≥ \underline{cd}(N) + s.

Using Lemma 6.3, we obtain

\underline{cd}(N) + s ≤ cd_{F[H]}(Λ)
≤ max\{\underline{\underline{cd}}(Λ), \underline{cd}(Λ)\}
≤ max\{\underline{\underline{cd}}(Λ), \underline{cd}(N) + 1\},

which implies that \underline{\underline{cd}}(Λ) ≥ \underline{cd}(N) + s. □

Example 6.5. In [31], Martínez-Pérez constructs, for any odd prime p, a Bestvina-Brady group H_{L_{p}} of type F and a semi-direct product Γ = H_{L_{p}} ⋊ C_{p} such that cd(H_{L_{p}}) = 3 but \underline{cd}(Γ) = 4. To put this into the context of Theorem C, we give a few details of this construction as presented in Example 3.6 of [31].

Consider the real projective plane ℝP^{2} with the standard CW-structure consisting of one cell in each dimension and let p be an odd prime. By a theorem of Jones (see [18, Theorem 1.1]), ℝP^{2} can be embedded in a contractible 3-dimensional finite C_{p}-CW-complex X such that X^{C_{p}} = ℝP^{2}. Let L be the flag triangulation of X and H_{L} the associated Bestvina-Brady group. Set Γ = H_{L} ⋊ C_{p} where the action of C_{p} on H_{L} is induced by the action of C_{p} on the flag complex L. Martínez-Pérez then obtains the cohomological finiteness conditions for Γ stated above.

Now, we let {p_{i} | i = 1, …, s} be a collection of distinct odd primes and for each i, denote by L_{i} the flag complex associated to the prime p_{i} in the construction of Martínez-Pérez. Note that the hypothesis of Theorem C are satisfied for q = 2 because n_{i} = 3 and L^{C_{p_{i}}} = ℝP^{2} for all i. Therefore, we can conclude that for each s ∈ ℕ, the group N_{s} = ∏_{i=1}^{s} H_{L_{i}} is an integral linear group of type F and there is a semi-direct product Γ_{s} = N_{s} ⋊ C_{d}, where d = ∏_{i=1}^{s} p_{i} such that

hd(N_{s}) = cd(N_{s}) = 3s and \underline{hd}(Γ_{s}) = \underline{cd}(Γ_{s}) = 4s.

Note that this implies

δ(N_{s}) − \underline{gd}(N_{s}) ≥ s

for every s ∈ ℕ. Moreover, taking any s ≥ 2, we obtain

gd(Λ_{s}) + s − 1 ≤ \underline{\underline{gd}}(Λ_{s}) < ∞

where Λ_{s} = N_{s} ⋊ ℤ an integral linear group of type F formed by taking the pull-back of Γ_{s} under the projection of ℤ onto C_{d}.

Remark 6.6. In [20, 46.4] (see also [14]), Lafont asked which finitely generated groups Γ have finite \underline{gd}(Γ) but infinite \underline{\underline{gd}}(Γ). Next, we explain how the invariant δ(N) is connected to this question.

Suppose there exists a countable group N such that \underline{gd}(N) < ∞ but δ(N) = ∞. This means that there is an infinite sequence of subgroups {K_{i}} of N and a collection of groups {Γ_{i}} where Γ_{i} is an extension of K_{i} by a finite cyclic group C_{d_{i}} such that

i ≤ \underline{cd}(Γ_{i}) < ∞,

for each i ∈ ℕ. Let Λ_{i} = K_{i} ⋊ ℤ be the pull-back of Γ_{i} under projection of ℤ onto C_{d_{i}} and let H_{i} be the infinite cyclic central subgroup of Λ_{i} such that Λ_{i}/H_{i} = Γ_{i} for each i ∈ ℕ. Proceeding as in the proof of Lemma 6.4, we obtain sup_{i∈ℕ}{cd_{F[H_{i}]}(Λ_{i})} = ∞. Since \underline{cd}(Λ_{i}) ≤ \underline{cd}(N) + 1 for each i, by Lemma 6.3, we have that sup_{i∈ℕ}\{\underline{\underline{cd}}(Λ_{i})\} = ∞.

Define Λ to be the infinite free product ∗_{i∈ℕ}Λ_{i}. Clearly, \underline{\underline{gd}}(Λ) = ∞. Since Λ acts on a tree with each vertex stabilizer conjugate to Λ_{i} for some i ∈ ℕ, using Corollary 4.1 of [6], it is not difficult to check that \underline{cd}(Λ) ≤ \underline{cd}(N) + 2. In the next proposition we will show that Λ can be embedded into a 2-generated group with the same Bredon cohomological dimension for the family of finite subgroups.

To summarize, we have just shown that if there exists a countable group N such that \underline{gd}(N) < ∞ but δ(N) = ∞. Then, there exists a 2-generated group Γ such that \underline{gd}(Γ) < ∞ but \underline{\underline{gd}}(Γ) = ∞.

Proposition 6.7. Let Γ be a countable group. Then Γ can be embedded into a 2-generated group H such that

\underline{cd}(Γ) ≤ \underline{cd}(H) ≤ max\{\underline{cd}(Γ), 2\}.

Proof. Following the proof of Higman-Neumann-Neumann in [15, Theorem IV], note that first, Γ is embedded in a group K which is a free product of Γ and ℤ. The Mayer-Vietoris sequence associated to the push-out construction of a model for \underline{E}K in Example 4.10 of [26] then implies that

\underline{cd}(K) ≤ max\{\underline{cd}(Γ), \underline{cd}(ℤ)\} = \underline{cd}(Γ).

Next, the group P is defined as the fundamental group of a graph of groups which is a (possibly infinite) wedge of circles with K as its vertex group and the edge subgroups all isomorphic to ℤ. Similarly, using a generalization of the construction in [26, 4.11] to a wedge of circles, one can deduce that \underline{cd}(P) ≤ max\{\underline{cd}(K), 2\}.

Now, the group Q is defined as an amalgamated free product of P and a free group on 2-generators. Again, by [26, 4.10] and the fact that non-abelian free groups have cohomological dimension 1, it follows that

\underline{cd}(Q) ≤ max\{\underline{cd}(P), 2\}.

In [15], it is also shown that Q is generated by the elements u, v, and b such that both u, b and b, v generate free subgroups of Q. Finally, the group H defined as

⟨Q, a | a^{−1}ba = u, a^{−1}va = b⟩

is 2-generated, and can be seen as an HNN-extension of Q relative to the isomorphism

θ : ⟨b, v⟩ → ⟨u, b⟩ : b ↦ u, v ↦ b.

Hence, by [26, 4.11], we have

\underline{cd}(H) ≤ max\{\underline{cd}(Q), 2\}.

The result now follows from combining the above inequalities. □

7. The Mayer-Vietoris sequence of Lück and Weiermann

Let F ⊆ H be two families of subgroups of a group G such that the set S = H ∖ F is equipped with a strong equivalence relation. Let [H] be the equivalence class represented by H ∈ S, denote the set of equivalence classes by [S] and let I be a complete set of representatives [H] of the orbits of the conjugation action of G on [S]. Lück and Weiermann obtained the following Mayer-Vietoris long exact sequence from the push-out diagram of Theorem 2.3. in [29].

Proposition 7.1 (Lück-Weiermann, [29]). Let M ∈ Mod-O_{H}G. There exists a long exact cohomology sequence

… → H_{H}^{i}(G, M) → ( ∏_{[H]∈I} H_{F[H]}^{i}(N_{G}[H], M)) ⊕ H_{F}^{i}(G, M) → ∏_{[H]∈I} H_{F∩N_{G}[H]}^{i}(N_{G}[H], M)

→ H_{H}^{i+1}(G, M) → ….

A consequence of this long exact sequence is the following algebraic analogue of Theorem 2.5.

Theorem 7.2. Let F ⊆ H be two families of subgroups of a group G such that S = H∖F is equipped with a strong equivalence relation. Denote the set of equivalence classes by [S] and let I be a complete set of representatives [H] of the orbits of the conjugation action of G on [S]. If there exists a natural number k such that for each [H] ∈ I

- cd_{F∩N_{G}[H]}(N_{G}[H]) ≤ k − 1;

- cd_{F[H]}(N_{G}[H]) ≤ k,

and such that cd_{F}(G) ≤ k, then cd_{H}(G) ≤ k.

Our goal in this section is to give an algebraic construction of the long exact sequence of Proposition 7.1. The added value of this approach is that our proof will produce long exact sequences for arbitrary Tor and Ext functors.

We start by recalling some basic facts about the (co)homology of an orbit category. We refer the reader to [24, section 9], [13] and [34] for more details.

Let O_{F}G be the orbit category. 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. We denote the morphism

φ : G/H → G/K : H ↦ xK

by G/H ⟶^{x} G/K. The set of morphisms from G/H to G/K is denoted by Mor(G/H, G/K).

A right O_{F}G-module is a contravariant functor M : O_{F}G → ℤ-mod. The category of right O_{F}G-modules is denoted by Mod-O_{F}G and is defined by the objects that are all the right O_{F}G-modules and the morphisms are all the natural transformations between the objects. A sequence

0 → M′ → M → M′′ → 0

in Mod-O_{F}G is called exact if it is exact after evaluating in G/H, for all H ∈ F. Take M ∈ Mod-O_{F}G and consider the left exact functor:

Hom_{F}(M, −) : Mod-O_{F}G → ℤ-mod : N ↦ Hom_{F}(M, N),

where Hom_{F}(M, N) is the abelian group of all natural transformations from M to N. By definition, M is a projective O_{F}G-module if and only if this functor is exact. It can be shown that Mod-O_{F}G is an abelian category that contains enough projective objects to construct projective resolutions. Hence, one can construct functors Ext_{O_{F} G}^{∗}(−, M) as the right derived functors of the contravariant right exact functor

Hom_{F}(−, M) : Mod-O_{F}G → ℤ-mod : N ↦ Hom_{F}(N, M).

These Ext functors have all the usual properties, and one has H_{F}^{∗}(G, M) = Ext_{O_{F} G}^{∗}(\underline{ℤ}, M) by definition.

Similarly, the category of left O_{F}G-modules is denoted by O_{F}G-Mod and is defined by the objects that are all covariant functors from O_{F}G to ℤ-mod and the morphisms are all the natural transformations between the objects.

For any L ∈ F, one can define the right O_{F}G-module

ℤ[−, G/L] : O_{F}G → ℤ-mod : G/H → ℤ[G/H, G/L],

where ℤ[G/H, G/L] is the free abelian group generated by Mor(G/H, G/L). Here, a morphism φ ∈ Mor(G/H_{1}, G/H_{2}) is mapped to

ℤ[G/H_{2}, G/L] → ℤ[G/H_{1}, G/L] : α ↦ α ∘ φ.

Similarly, one can define the left O_{F}G-module ℤ[G/L, −] for every L ∈ F.

Given M ∈ Mod-O_{F}G and N ∈ O_{F}G-Mod, one defines their tensor product M ⊗_{O_{F} G}N to be the following abelian group

M ⊗_{O_{F} G} N = ⨁_{S∈F} M(G/S) ⊗ N(G/S)/I,

where I is the abelian group generated by the elements M(φ)(m) ⊗ n − m ⊗ N(φ)(n), for all φ ∈ Mor(G/H, G/K), m ∈ M(G/K), n ∈ N(G/H) and for all H, K ∈ F. We view elements of M ⊗_{O_{F} G} N as finite sum of elementary tensors of the form m ⊗ n ∈ M(G/H) ⊗ N(G/H) for all H ∈ F, that satisfy M(φ)(m) ⊗ n = m ⊗ N(φ)(n) for all φ ∈ Mor(G/H, G/K). When the group is clear from the context, we will denote M⊗_{O_{F} G}N by M ⊗_{F} N.

Given N ∈ O_{F}G-Mod, define the right exact functor

− ⊗_{O_{F} G} N : Mod-O_{F}G → ℤ-mod : M → M ⊗_{O_{F} G} N.

The functors Tor_{∗}^{O_{F} G}(−, N) are the left derived functors of this right exact functor. These Tor functors can be computed using projective resolutions, or more generally, using flat resolutions. The n-th Bredon homology of the group G with coefficients in N ∈ O_{F}G-Mod is by definition

H_{∗}^{F}(G, N) = Tor_{∗}^{O_{F} G}(\underline{ℤ}, N).

There is a notion of Bredon homological dimension of the group G for the family F, which is defined as the integer

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

In general, the inequality hd_{F}(G) ≤ cd_{F}(G) always holds. For countable groups G, one even has

hd_{F}(G) ≤ cd_{F}(G) ≤ hd_{F}(G) + 1.

Let K be a group and let V be a family of subgroups of K. Associated to a functor π : O_{V}K → O_{F}G, one has a restriction functor

res_{π} : Mod-O_{F}G → Mod-O_{V}K : M ↦ M ∘ π

and an induction functor

ind_{π} : Mod-O_{V}K → Mod-O_{F}G : M ↦ M(?) ⊗_{O_{V}K} ℤ[−, π(?)].

The functor ind_{π} is left adjoint to the functor res_{π}. In particular, there is an isomorphism

(6)  Hom_{O_{F} G}(ind_{π}(M), N) ≅ Hom_{O_{V}K}(M, res_{π}(N))

that is natural in M ∈ Mod-O_{V}K and N ∈ Mod-O_{F}G.

Convention. From now on, restriction functors will be omitted from the notation.

Before we proceed, let us establish some notation. The induction functor associated to the functor

O_{F}G → O_{H}G : G/F ↦ G/F

will be denoted by ind_{F}^{H}. The induction functor associated to the functor

O_{F∩N_{G}[H]}N_{G}[H] → O_{H}G : N_{G}[H]/F ↦ G/F

will be denoted by ind_{F∩N_{G}[H]}^{H}. And finally, the induction functor associated to the functor

O_{F[H]}N_{G}[H] → O_{H}G : N_{G}[H]/F ↦ G/F

will be denoted by ind_{F[H]}^{H}.

In the next three lemmas, we will prove some elementary properties of the induction functors introduced above.

Lemma 7.3. Let G be a group and let F ⊆ H be an inclusion of families of subgroups of G. If M ∈ Mod-O_{F}G, then M and ind_{F}^{H}(M) are isomorphic as O_{F}G-modules and ind_{F}^{H}(M)(G/S) = 0 for all S ∈ H ∖ F.

Proof. Let H ∈ F. Note that in M ⊗_{O_{F} G} ℤ[G/H, −], we have m⊗φ = M(φ)(m)⊗Id_{H} for any m ∈ M(G/K) and φ ∈ Mor(G/H, G/K). Here Id_{H} denotes the identity morphism G/H → G/H. Using this observation, one checks that the map

η(H) : M ⊗_{O_{F} G} ℤ[G/H, −] → M(G/H),

that sends m ⊗ φ ∈ M(G/K) ⊗ ℤ[G/H, G/K] to M(φ)(m) ∈ M(G/H), is a well-defined isomorphism. Moreover, it is easily seen that the maps η(H) assemble to form a natural transformation η : ind_{F}^{H}(M) → M of O_{F}G-modules. This proves the first statement of the lemma.

Now, let S ∈ H ∖ F. By definition, we have

ind_{F}^{H}(M)(G/S) = M(−) ⊗_{F} ℤ[G/S, −].

If there existed a subgroup F ∈ F and a morphism G/S ⟶^{x} G/F, then this would imply that S ∈ F because F is subgroup and conjugation closed. We conclude that M(−) ⊗_{F} ℤ[G/S, −] = 0. This completes the proof. □

Lemma 7.4. Let M ∈ Mod-O_{H}G, H ∈ S and K ∈ F. The natural map

ind_{F∩N_{G}[H]}^{H}(M)(G/K) → ind_{F[H]}^{H}(M)(G/K) : m ⊗ φ ↦ m ⊗ φ

is an isomorphism.

Proof. Let L ∈ F[H], m ∈ M(G/L) and (when possible) φ : G/K ⟶^{x} G/L in ℤ[G/K, G/L]. We have

(G/K ⟶^{x} G/L) = (G/K ⟶^{x} G/K^{x} → G/L)

and K^{x} ∈ F ∩ N_{G}[H]. This implies that

m ⊗ φ = M(G/K^{x} → G/L)(m) ⊗ (G/K ⟶^{x} G/K^{x}) ∈ M ⊗_{F[H]} ℤ[G/K, −].

This shows that every element of M ⊗_{F[H]} ℤ[G/K, −] is a sum of elementary tensors of the form n ⊗ (G/K ⟶^{x} G/K^{x}) for some x ∈ G and some n ∈ M(G/K^{x}). Using this fact, it is not difficult to check that the map under consideration is an isomorphism. □

Lemma 7.5. The induction functors ind_{F}^{H}, ind_{F∩N_{G}[H]}^{H} and ind_{F[H]}^{H} are exact and preserve projectives.

Proof. All induction functors preserve projectives since they are left adjoint to a restriction functor which is exact.

Let Γ be a group and let F ⊆ H be two families of subgroups of Γ. Then the induction functor associated to the functor

O_{F}Γ → O_{H}Γ : Γ/F ↦ Γ/F

is exact, by Lemma 7.3. In particular, ind_{F}^{H} is exact. Now, let Γ be a group, K be a subgroup of Γ and let F be a family of subgroups of Γ. Lemma 2.9 in [37] (or see Proposition 3.26 in [13]) shows that the induction functor associated to the functor

O_{F∩K}K → O_{F}Γ : K/F ↦ Γ/F

is exact. Since the functors ind_{F∩N_{G}[H]}^{H} and ind_{F[H]}^{H} can be written as a composition of the two previous induction functors, we conclude that they are exact as well. □

The following proposition provides the main algebraic tool needed to construct the long exact Mayer-Vietoris sequence.

Proposition 7.6. Let M ∈ Mod-O_{H}G. We have a short exact sequence

0 → ⨁_{[H]∈I} ind_{F∩N_{G}[H]}^{H}(M) → ( ⨁_{[H]∈I} ind_{F[H]}^{H}(M)) ⊕ ind_{F}^{H}(M) → M → 0.

of O_{H}G-modules.

Proof. First, notice that

ind_{F∩N_{G}[H]}^{H}(M)(K) = ind_{F}^{H}(M)(K) = 0

if K ∈ S and that ind_{F}^{H}(M)(K) = M(K) if K ∈ F, by Lemma 7.3. Suppose K ∈ F. Let us first define the map

i(K) : ⨁_{[H]∈I} ind_{F∩N_{G}[H]}^{H}(M)(K) → ( ⨁_{[H]∈I} ind_{F[H]}^{H}(M)(K)) ⊕ M(G/K).

Let

m ⊗ φ ∈ ind_{F∩N_{G}[H]}^{H}(M)(K) = M ⊗_{F∩N_{G}[H]} ℤ[G/K, −]

and define

i(K)(m ⊗ φ) = (m ⊗ φ, −M(φ)(m)) ∈ ind_{F[H]}^{H}(M)(K) ⊕ M(G/K).

If K ∈ S, then set i(K) = 0. One can now easily check that the maps i(K), for K ∈ H, assemble to form a natural transformation

i : ⨁_{[H]∈I} ind_{F∩N_{G}[H]}^{H}(M) → ( ⨁_{[H]∈I} ind_{F[H]}^{H}(M)(K)) ⊕ ind_{F}^{H}(M).

Let K ∈ F and define the map

p(K) : ( ⨁_{[H]∈I} ind_{F[H]}^{H}(M)(G/K)) ⊕ M(G/K) → M(G/K) : (m ⊗ φ, n) ↦ M(φ)(m) + n.

If K ∈ S, then define

p(K) : ( ⨁_{[H]∈I} ind_{F[H]}^{H}(M)(G/K)) → M(G/K) : m ⊗ φ ↦ M(φ)(m).

One can check that the maps p(K) assemble to form a natural transformation

p : ( ⨁_{[H]∈I} ind_{F[H]}^{H}(M)(G/K)) ⊕ ind_{F}^{H}(M) → M.

We claim that the sequence

0 → ⨁_{[H]∈I} ind_{F∩N_{G}[H]}^{H}(M) ⟶^{i} ( ⨁_{[H]∈I} ind_{F[H]}^{H}(M)) ⊕ ind_{F}^{H}(M) ⟶^{p} M → 0

is exact. To prove this, first let K ∈ F. Then p(K) is obviously surjective and the injectivity of i(K) follows from Lemma 7.4. Clearly, we also have p(K) ∘ i(K) = 0. If an element of

( ⨁_{[H]∈I} ind_{F[H]}^{H}(M)(K)) ⊕ M(G/K)

is in the kernel of p(K), then Lemma 7.4 implies that this element is in the image if i(K). Now let K ∈ S. We need to show that

(7)  p(K) : ⨁_{[H]∈I} M ⊗_{F[H]} ℤ[G/K, −] → M(G/K)

is an isomorphism. Consider [H] ∈ I and assume that there exists an L ∈ F[H] such that ℤ[G/K, G/L] is non-zero. Then there exists an x ∈ G such that K^{x} ⊆ L. Since L ∼ H, it follows that K ∼ ^{x}H. By definition of I, there can be at most one [H] ∈ I such that ℤ[G/K, G/L] is non-zero for some L ∈ F[H]. On the other hand, there must exist an [S] ∈ I such that K^{x} ∼ S. This implies that K^{x} ∈ F[S], so ℤ[G/K, G/K^{x}] is non-zero. We conclude that there exists a unique [S] ∈ I such that (7) transforms to

p(K) : M ⊗_{F[S]} ℤ[G/K, −] → M(G/K) : m ⊗ φ ↦ M(φ)(m),

and such that K^{x} ∈ F[S] for some x ∈ G. Denote the morphism G/K ⟶^{x} G/K^{x} by φ_{0}. Let L ∈ F[S] ∖ F ∩ N_{G}[S] and consider (when possible) a morphism φ : G/K ⟶^{y} G/L. The existence of this morphism implies that K ∼ ^{y}S. Since we also have K ∼ ^{x}S, it follows that x^{−1}y ∈ N_{G}[S]. By noting that

(G/K ⟶^{y} G/L) = (G/K ⟶^{x} G/K^{x} ⟶^{x^{−1}y} G/L),

we see that

m ⊗ φ = M(G/K^{x} ⟶^{x^{−1}y} G/L)(m) ⊗ φ_{0} ∈ M ⊗_{F[S]} ℤ[G/K, −],

for any m ∈ M(G/L). This shows that every element in M ⊗_{F[S]} ℤ[G/K, −] is of the form m ⊗ φ_{0} for some m ∈ M(G/K^{x}). Since the morphism φ_{0} : G/K ⟶^{x} G/K^{x} has a two-sided inverse, the map M(φ_{0}) is an isomorphism. It follows that p(K) is an isomorphism. □

We can now prove the main result of this section.

Theorem 7.7. Let F ⊆ H be two families of subgroups of a group G such that the set S = H ∖ F is equipped with a strong equivalence relation. Let [H] be the equivalence class represented by H ∈ S, denote the set of equivalence classes by [S] and let I be a complete set of representatives [H] of the orbits of the conjugation action of G on [S]. Take M, N ∈ Mod-O_{H}G and T ∈ O_{H}G-Mod. Then, there exists a long exact sequence of Ext functors

… → Ext_{O_{H}G}^{i}(M, N) → ( ∏_{[H]∈I} Ext_{O_{F[H]}N_{G}[H]}^{i}(M, N)) ⊕ Ext_{O_{F} G}^{i}(M, N) →

∏_{[H]∈I} Ext_{O_{F∩N_{G}[H]}N_{G}[H]}^{i}(M, N) → Ext_{O_{H}G}^{i+1}(M, N) → …,

and a long exact sequence of Tor functors

… → ⨁_{[H]∈I} Tor_{i}^{O_{F∩N_{G}[H]}N_{G}[H]}(M, T) → ( ⨁_{[H]∈I} Tor_{i}^{O_{F[H]}N_{G}[H]}(M, T)) ⊕ Tor_{i}^{O_{F} G}(N, T) →

Tor_{i}^{O_{H}G}(M, T) → ⨁_{[H]∈I} Tor_{i−1}^{O_{F∩N_{G}[H]}N_{G}[H]}(M, T) → ….

Proof. Using Lemma 7.5 and the adjointness between induction and restriction (6), one checks that the desired long exact sequences are the long exact sequences induced by Ext_{O_{H}G}^{∗}(−, N) and Tor_{∗}^{O_{H}G}(−, T) from the short exact sequence of coefficients of Proposition 7.6 applied to M. □

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Department of Mathematics, KU Leuven, Kortrijk, Belgium

E-mail address: Dieter.Degrijse@kuleuven-kulak.be

Department of Mathematics, KU Leuven, Kortrijk, Belgium

E-mail address: Nansen.Petrosyan@kuleuven-kulak.be