JUMPS IN COHOMOLOGY AND FREE GROUP ACTIONS

NANSEN PETROSYAN

Abstract. A discrete group G has periodic cohomology over R if there is an element in a cohomology group, cup product with which induces an isomorphism in cohomology after a certain dimension. Adem and Smith showed if R = ℤ, then this condition is equivalent to the existence of a finite dimensional free-G-CW-complex homotopy equivalent to a sphere. It has been conjectured by Olympia Talelli, that if G is also torsion-free then it must have finite cohomological dimension. In this paper we use the implied condition of jump cohomology over R to prove the conjecture for HF-groups and solvable groups. We also find necessary conditions for free and proper group actions on finite dimensional complexes homotopy equivalent to closed, orientable manifolds.

1. Introduction

Definition 1.1. A discrete group G has periodic cohomology over R if there is a cohomology class α ∈ Ext_{RG}^{∗}(R, R) with |α| > 0 and an integer n ≥ 0 such that the cup product map

α ∪ − : Ext_{RG}^{i}(R, M) → Ext_{RG}^{i+|α|}(R, M)

is an isomorphism for every RG-module M and every integer i ≥ n.

If R = ℤ, then we simply say that G has periodic cohomology. In this case Adem and Smith [1] proved that this condition is equivalent to the existence of a finite dimensional free-G-CW-complex homotopy equivalent to a sphere. If a group G has periodic cohomology then it has periodic cohomology over any commutative ring R with a unit. This definition of periodicity using the cup product differs from the general definition of periodic cohomology of a group. Namely, a group G has periodic cohomology after k-steps if there exists an integer q ≥ 0 such that the functors H^{i}(G, −) and H^{i+q}(G, −) are naturally equivalent for all i > k. We will call this classical periodic cohomology. It is immediate that the cup product notion

Date: June 22, 2012.

Key words and phrases. spectral sequence, group cohomology.

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of periodic cohomology implies the classical periodic cohomology. It is a conjecture by O. Talelli that the two notions are the same. She has proved that the conjecture holds in the case of the HF-groups introduced by P. Kropholler in [7]. This is the smallest class of groups which contains all finite groups and which contains all groups G, whenever G acts cellularly on a finite dimensional contractible CW-complex with all isotropy subgroups already in HF. This is a large class of groups. Among others, it contains all countable linear groups, all countable solvable groups and all groups with finite virtual cohomological dimension.

The following was conjectured by Talelli in [11], Conjecture III:

Conjecture (O. Talelli). Every torsion-free discrete group G with classical periodic cohomology after some steps has finite cohomological dimension.

The Conjecture was shown to be true for the class of HF-groups by Mislin and Talelli in [8]. In our paper we investigate a weaker condition, implied by cohomological periodicity, for a discrete group G.

Definition 1.2. A discrete group G has jump cohomology over R if there exists an integer k ≥ 0, such that if H is any subgroup of G with cd_{R}(H) < ∞, then cd_{R}(H) ≤ k. The bound k will be called the jump height over R. If R = ℤ then we say G has jump cohomology and a jump height k.

By naturality of the cup product it follows that if a group G has periodic cohomology over R, then G has jump cohomology over R. In section 2 we prove the following proposition which relates the geometric aspects of the notion to the jump cohomology.

Proposition 1.3. Let G be a discrete group that acts freely and properly on an edimensional CW-complex E. Suppose there is an integer n ≤ e such that H_{i}(E, ℤ) = 0 for all i > n, and H_{n}(E, ℤ) ≅ ℤ. Set k = e − n. If H is any subgroup of G with cd(H) < ∞ (hd(H) < ∞), then cd(H) ≤ k (respectively hd(H) ≤ k).

In section 3 we investigate the classes of HF-groups and solvable groups without R-torsion. We prove the following two theorems.

Theorem 1.4. Let G be a discrete group without R-torsion with jump cohomology of height k over R. If G is an HF-group, then cd_{R}(G) ≤ k. In particular, any HF-group G has jump cohomology of height k over ℚ if and only if cd_{ℚ}(G) ≤ k.

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Theorem 1.5. Let G be a solvable group that acts freely and properly on an edimensional CW-complex E. Suppose there is an integer n ≤ e such that H_{i}(E, ℤ) = 0 for all i > n, and H_{n}(E, ℤ) ≅ ℤ, then h(G) ≤ e − n. If G is also R-torsion-free, hd_{R}(G) ≤ e − n.

Our results give affirmative answers for the case of HF-groups, and when R is a domain of characteristic zero for the case of solvable groups to the conjecture we believe is a natural generalization of the conjecture by Talelli.

Conjecture 1.6. For every discrete group G without R-torsion the following are equivalent.

(1) G has jump cohomology of height k over R.

(2) G has periodic cohomology over R starting in dimension k + 1.

(3) cd_{R}(G) ≤ k

This conjecture in particular would imply Talelli’s Conjecture. It also says that every group with periodic cohomology over ℚ must have cd_{ℚ}(G) < ∞. Periodic or jump cohomology is a rather strong condition for the cohomology of G. We present the example of the Thompson group F = ⟨x_{0}, x_{1}, x_{2}, ...|x_{i}^{−1}x_{n}x_{i} = x_{n+1} for all i < n and n ∈ ℕ⟩, which is of type FP_{∞}. It has periodic cohomology with integral coefficients and vanishing cohomology with ℤF -module coefficients, but it does not have jump cohomology over any ring R.

Next we apply Theorem 1.5 to groups that can act freely and properly on a finite dimension complex homotopy equivalent a closed, orientable manifold.

Corollary 1.7. If G is a solvable group, that acts freely and properly on an (m+n)- dimensional complex homotopy equivalent to closed, orientable manifold of dimension n, then the Hirsch rank h(G) ≤ m.

Corollary 1.8. If G has a non-countable, torsion-free, solvable subgroup, then G cannot act freely and properly on a finite dimensional complex homotopy equivalent to a closed, orientable manifold.

Lastly, in section 4, we find an obstruction for a countable group acting on certain complexes to be free. We apply the next proposition to groups that admit free and proper actions on finite dimension complexes homotopy equivalent to closed, orientable manifolds.

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Proposition 1.9. Let G be a finitely generated torsion-free group that acts freely and properly on an n-dimensional CW-complex X such that the G-invariant submodule (H_{n−1}(X, R))^{G} ≠ 0 . Then G is a free group if and only if Ext_{RG}^{2}(R, H_{n}(X, R)) is trivial.

Theorem 1.10. If G is a torsion-free countable group that acts freely and properly on an n-dimensional complex homotopy equivalent to a closed, orientable manifold of dimension n − 1, then G must be free.

I would like to thank Alejandro Adem, Donald Passman and Olympia Talelli for their conversations and advice. Alejandro Adem and Olympia Talelli read an earlier version of the paper and made useful suggestions.

2. Periodicity and Boundedness

Unless otherwise specified, R will denote any commutative ring with a unit.

A group G is said to be R-torsion-free if for every finite subgroup H of G, n · 1_{R} is an invertible element in R, where n is the order of H. One should observe that this is equivalent to the following definition, which will be used in Lemma 3.4.

G is R-torsion-free if and only if for each prime p, such that there exists a nonidentity element g ∈ G with g^{p} = 1, p · 1_{R} is an invertible element in R.

Definition 2.1. Cohomological dimension of a discrete group G over R, denoted cd_{R}(G), is defined as

inf{n : Ext_{RG}^{i}(R, −) = 0 for i > n}

and the homological dimension, hd_{R}(G), of G over R is inf{n : Tor_{i}^{RG}(R, −) = 0 for i > n}

An immediate consequence of G having finite cohomological or homological dimension is that G must be R-torsion-free.

Definition 2.2. A discrete group G has periodic cohomology over R if there is a cohomology class α ∈ Ext_{RG}^{∗}(R, R) with |α| > 0 and an integer n ≥ 0 such that the cup product map

α ∪ − : Ext_{RG}^{i}(R, M) → Ext_{RG}^{i+|α|}(R, M)

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is an isomorphism for every RG-module M and every integer i ≥ n. The smallest degree of such α will be the period of the periodic cohomology of G. If R = ℤ, we simply say that G has periodic cohomology.

The above notion of periodicity was used by Adem and Smith in [1]. They prove a generalization of Wall’s conjecture [8] assuming that the periodicity is induced by the cup product with a cohomology class in H^{∗}(G, ℤ).

Theorem 2.3. (Adem-Smith). A discrete group G has periodic cohomology if and only if G acts freely and properly on a finite dimensional complex homotopy equivalent to a sphere.

Let us investigate a weaker cohomological condition, which arises naturally, when a group G has a periodic cohomology over a ring R. It will be our main interest in section 3, where we will investigate specific types of groups satisfying the condition.

Definition 2.4. A discrete group G has jump cohomology (homology) over R if there exists an integer k ≥ 0, such that if H is any subgroup of G with cd_{R}(H) < ∞ (hd_{R}(H) < ∞), then cd_{R}(H) ≤ k (hd_{R}(H) ≤ k). The bound k will be called the jump height over R. If R = ℤ then we simply say G has jump cohomology (homology) and a jump height k.

This next proposition gives a more geometric meaning to the jump height.

Proposition 2.5. Let G be a discrete group acting freely and properly on an edimensional CW-complex E. Suppose there is an integer n ≤ e such that H_{i}(E, ℤ) = 0 for all i > n, and H_{n}(E, ℤ) ≅ ℤ. Set k = e − n. Then G has jump cohomology and homology of height k.

Proof. We will show the proof of the proposition concerning cohomology. The proof of the homological version is similar.

Suppose H is a subgroup of G with finite cohomological dimension m. We can assume that the induced action of H on H_{n}(E, ℤ) is trivial, otherwise we can pass to an index two subgroup of H and use Serre’s Theorem, [9].

We get the following fibration

E ⟶ E ×_{H} EH⟶^{π}BH

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where E ×_{H} EH is the Borel Construction and it is homotopy equivalent to E/H. Therefore, by the Leray-Serre spectral sequence we have

E_{2}^{p,q} = H^{p}(BH, H^{q}(E, M)) ⟹ H^{p+q}(E/H, M)

There exists a ℤG-module F such that H^{m}(H, F) ≠ 0. By the usual corner argument of the sequence it follows

H^{m+n}(E/H, F) ≅ H^{m}(BH, H^{n}(E, F))

By the Universal Coefficient Theorem we have the following isomorphism of ℤHmodules

H^{n}(E, F) ≅ Hom(H_{n}(E, ℤ), F) ⊕ N ≅ F ⊕ N

where N = Ext(H_{n−1}(E, ℤ), F). This shows that

H^{m+n}(E/H, F) ≅ H^{m}(BH, F) ⊕ H^{m}(BH, N)

Therefore, H^{m+n}(E/H, F) ≠ 0 and m + n ≤ e.

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Remark 2.6. If there exist a finite dimensional free-G-CW-complex E homotopy equivalent to a sphere S^{n}, then G has a periodic cohomology over any ring R. For we can always construct the join E ∗ E with the diagonal G-action. This complex is homotopy equivalent to S^{2n+1} and H_{∗}(E ∗ E, ℤ) has the trivial G-action. Therefore the induced G-action on H^{∗}(E ∗E, R) is also trivial. Then the Gysin exact sequence of cohomology over R shows that the cup product with the Euler class of the spherical fibration induces an isomorphism in cohomology of G after dimension 2k, where 2k is the jump height over R. This, together with Theorem 2.3, yields that, if a group G has periodic cohomology, then it has periodic cohomology over any ring R.

Now, suppose G has periodic cohomology over R starting in dimension k +1. If H is any subgroup of G with finite cohomological dimension h over R, using Shapiro’s Lemma, we have

Ext_{RH}^{k+1}(R, M) ≅ Ext_{RG}^{k+1}(R, Coind_{H}^{G}M) ⟶^{α^{h}∪−}

Ext_{RG}^{k+h|α|+1}(R, Coind_{H}^{G}M) ≅ Ext_{RH}^{k+h|α|+1}(R, M) = 0

for any RH-module M. This establishes the following lemma.

Lemma 2.7. If a group G has periodic cohomology over R starting in dimension k + 1, then G has a jump cohomology of height k over R.

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Therefore, among the three cohomological conditions stated in the conjecture, jump cohomology is the weakest.

3. Some classes of groups

The following class of groups was introduced by P. Kropholler in [7]. Our next theorem shows that they satisfy the conjecture stated in section 1.

Definition 3.1. Let X denote a class of groups. Define HX to be the smallest class of groups containing X with the property: if a group G acts cellularly on a finite dimensional contractible CW-complex with all isotropy subgroups in HX, then G is in HX.

In [7] many properties of these classes of groups, such as subgroup and extension closure, closure under countable direct unions and free product, were shown. The main interest to us is the hierarchical description of HX -groups defined by operations H_{α} for each ordinal α inductively: -H_{0}X = X For each β > 0, -H_{β}X is the class of groups G which act cellularly on a finite dimensional contractible CW-complex X such that for each cell σ of X the isotropy group G_{σ} is in H_{α}X for some α < β.

It is immediate that a group G is a HX -group if and only if there is an α such that G is in H_{α}X. Let F denote the class of all finite groups. We obtain the following result pertaining to HF-groups.

Theorem 3.2. Let G be a discrete group without R-torsion with jump cohomology of height k over R. If G is an HF-group, then cd_{R}(G) ≤ k. In particular, any HF-group G has jump cohomology of height k over ℚ if and only if cd_{ℚ}(G) ≤ k.

First we need a lemma.

Lemma 3.3. If a discrete group G acts cellularly on an n-dimensional contractible complex X, such that for any cell stabilizers of X, cd_{R}(G_{σ}) ≤ k, then cd_{R}(G) ≤ k+n.

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Proof. By considering the double complex Hom_{RG}(P_{∗}, C_{R}^{∗}(X, M)), where P_{∗} is a projective resolution of R over RG and C_{R}^{∗}(X, M) is the cellular cochain complex of X with RG-module coefficients M (see for example [3]) we can derive the firstquadrant spectral sequence

E_{1}^{p,q} = Ext_{RG}^{q}(R, C_{R}^{p}(X, M)) ⟹ Ext_{RG}^{p+q}(R, C_{R}^{∗}(X, M))

where Ext_{RG}^{∗}(R, C_{R}^{∗}(X, M)) is the cohomology of the total complex associated to the double complex.

For each p-cell σ of X there exists a G_{σ}-module R_{σ}. This module is isomorphic to R additively, and G_{σ} acts on it through the orientation character. Let M^{σ} = Hom_{R}(R_{σ}, M). Let X_{p} denote the collection of all the p-cells and let Σ_{p} be a set of representatives of all the G-orbits in X_{p}. We have the following decomposition

C_{R}^{p}(X, M) = Hom_{R}(C_{p}^{R}(X), M) = ⨁_{σ∈X_{p}}Hom_{R}(R_{σ}, M) =

⨁_{σ∈Σ_{p}}Hom_{R}(Ind_{G_{σ}}^{G}R_{σ}, M) = ⨁_{σ∈Σ_{p}}Coind_{G_{σ}}^{G}M^{σ}

Now, by Shapiro’s lemma,

Ext_{RG}^{q}(R, C_{R}^{p}(X, M)) ≅ ⨁_{σ∈Σ_{p}}Ext_{R[G_{σ}]}^{q}(R, M^{σ}).

Since X is contractible the two cochain complexes C_{R}^{∗}(X, M) and C_{R}^{∗}(pt., M) are homotopy equivalent. Hence,

Ext_{RG}^{∗}(R, C_{R}^{∗}(X, M)) ≅ Ext_{RG}^{∗}(R, C_{R}^{∗}(pt., M)) = Ext_{RG}^{∗}(R, M).

The spectral sequence then becomes

E_{1}^{p,q} = ⨁_{σ∈Σ_{p}}Ext_{R[G_{σ}]}^{q}(R, M^{σ}) ⟹ Ext_{RG}^{p+q}(R, M)

It follows that E_{1}^{p,q} = 0 if p > n or q > k. We can infer inductively for i = 0, 1, 2, ... that the differentials d_{i}^{p,q} = 0 and the terms of the spectral sequence E_{i}^{p,q} = 0 if p > n or q > k. Therefore, E_{∞}^{p,q} ≅ E_{1}^{p,q} = 0 if p > n or q > k and Ext_{RG}^{p+q}(R, M) = 0 if p > n and q > k. Thus cd_{R}(G) ≤ n + k.

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Proof of Theorem. We will show that cd_{R}(G) ≤ k by transfinite induction, using the hierarchical description of HF-groups.

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If G is in H_{0}F = F, then G is a finite group. So, cd_{R}(G) = 0. Assume now that for a fixed β, and all α < β, cd_{R}(H) ≤ k for any subgroup H of G in H_{α}F. Suppose G is an H_{β}F-group. By assumption there is a finite dimensional contractible G-CW-complex X such that for any cell σ of X the isotropy group G_{σ} is an H_{α}F-group for some α < β. So, by induction, cd(G_{σ}) ≤ k. Let n be the dimension of X. Now, by the lemma, cd_{R}(G) ≤ k + n. Hence, by hypothesis, cd_{R}(G) ≤ k.

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Our next goal is to study solvable groups with jump cohomology. For this we need to assume R is an integral domain. Note that by the previous theorem the conjecture stated in section 1 is true for countable R-torsion-free solvable groups, for they are HF-groups, for an arbitrary commutative ring R. What proceeds extends this to uncountable solvable groups when R is a domain. In fact, our arguments show that if a torsion-free solvable group has jump cohomology, then it is countable, since it must have finite Hirsch rank. We denote by 𝔽 the fraction field of R, so 𝔽 is a field of an arbitrary characteristic.

Lemma 3.4. Let 1 = G_{0} ⊂ G_{1} ⊂ ... ⊂ G_{n} = G be an upper central series of G. If G is an R-torsion-free group, then so are the factor groups G_{i}/G_{i−1}, i = 1, .., n.

Proof. Recall that an upper central series for G can be defined inductively by letting G_{i} to be the normal subgroup of G such that G_{i}/G_{i−1} = Z(G/G_{i−1}), where Z(G/G_{i−1}) is the center of the quotient group G/G_{i−1}.

Let us proceed by induction. Suppose, for all j ≤ i, G_{j}/G_{j−1} are R-torsion-free. We need to show that the same is true for the quotient group G_{i+1}/G_{i}. Suppose x̄ ∈ G_{i+1}/G_{i} such that x̄^{p} = 1 for a prime number p, and p·1_{R} is not invertible in the ring R. This shows that x^{p} ∈ G_{i}. Now for any element g of G, [g, x]^{p} ≡ [g, x^{p}] ≡ 1 mod G_{i−1}. Since [g, x] ∈ G_{i}, the induction shows [g, x] ∈ G_{i−1}. So, in the quotient group G/G_{i−1}, [ḡ, x̄] = 1 for all ḡ ∈ G/G_{i−1}. Therefore, by definition, x̄ ∈ G_{i}/G_{i−1} and x ∈ G_{i}. It follows that x̄ is the identity element in G_{i+1}/G_{i}.

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Proposition 3.5. Let G be a nilpotent group without R-torsion with jump homology of height k over R. Then hd_{R}(G) = h(G) ≤ k and Tor_{h(G)}^{RG}(R, 𝔽) ≅ 𝔽, where h(G) is the Hirsch rank of G.

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Proof. Let 1 = G_{0} ⊂ G_{1} ⊂ ... ⊂ G_{n} = G be a central series of G with R-torsion-free factor groups. We use induction on n.

Let us assume h(G_{i}) = hd_{R}(G_{i}) = h_{i}, and Tor_{h_{i}}^{RG_{i}}(R, 𝔽) ≅ 𝔽^{ω_{i}}, where 𝔽^{ω_{i}} is additively isomorphic to 𝔽 with a trivial G/G_{i}-action. Let A ⊂ G_{i+1}/G_{i} and A ≅ ℤ^{m}. Let G′_{i+1} = φ^{−1}(A), where φ : G_{i+1} → G_{i+1}/G_{i} is the canonical quotient map. So, there exists an extension, 1 → G_{i} → G′_{i+1} → A → 1.

By the corner argument of the Lyndon-Hochschild-Serre spectral sequence of the extension above it follows,

Tor_{m+h_{i}}^{RG′_{i+1}}(R, 𝔽) ≅ Tor_{m}^{RA}(R, Tor_{h_{i}}^{RG_{i}}(R, 𝔽)) ≅ Tor_{m}^{RA}(R, 𝔽^{ω_{i}})

Since A ≅ ℤ^{m},

Tor_{m}^{RA}(R, 𝔽^{ω_{i}}) ≅ ⋀^{m}(A ⊗ 𝔽^{ω_{i}}) ≅ ⋀^{m}(ℤ^{m} ⊗ 𝔽^{ω_{i}}) ≅ 𝔽

where ⋀^{∗}(A ⊗ 𝔽^{ω_{i}}) is the exterior algebra of the 𝔽-vector space A ⊗ 𝔽^{ω_{i}}. This shows Tor_{m+h_{i}}^{RG′_{i+1}}(R, 𝔽) ≠ 0. On the other hand, since hd_{R}(G_{i}) = h_{i} and hd_{R}(A) = m, the corner argument of the spectral sequence also shows, hd_{R}(G′_{i+1}) ≤ m + h_{i}. Thus hd_{R}(G′_{i+1}) = m + h_{i}. So, by the hypothesis m + h_{i} ≤ k, showing m ≤ k. This proves that G_{i+1}/G_{i} is a finite-rank abelian group. Thus, G_{i+1} has finite Hirsch rank, and since G_{i+1}/G_{i} is also R-torsion-free, h(G_{i+1}/G_{i}) = hd_{R}(G_{i+1}/G_{i}). So, again by the corner argument of the spectral sequence associated to the extension, 1 → G_{i} → G_{i+1} → G_{i+1}/G_{i} → 1, we have hd_{R}(G_{i+1}) ≤ h(G_{i+1}).

Let m = hd_{R}(G_{i+1}/G_{i}). The spectral sequence gives

Tor_{m+h_{i}}^{RG_{i+1}}(R, 𝔽) ≅ Tor_{m}^{R[G_{i+1}/G_{i}]}(R, Tor_{h_{i}}^{RG_{i}}(R, 𝔽)) ≅
Tor_{m}^{R[G_{i+1}/G_{i}]}(R, 𝔽^{ω_{i}}) ≅ ⋀^{m}(G_{i+1}/G_{i} ⊗ 𝔽^{ω_{i}})

The last isomorphism follows from the fact that G_{i+1}/G_{i} is an R-torsion-free abelian group and 𝔽^{ω_{i}} has trivial G_{i+1}/G_{i}-action. Also, since G_{i+1}/G_{i} is central in G/G_{i}, the induced action of G/G_{i+1} on ⋀^{m}(G_{i+1}/G_{i} ⊗ 𝔽^{ω_{i}}) is trivial. So the module ⋀^{m}(G_{i+1}/G_{i} ⊗ 𝔽^{ω_{i}}) is 𝔽^{ω_{i+1}}, and hd_{R}(G_{i+1}) = h(G_{i+1}). This completes the induction.

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Our next result requires a version of a theorem of Stammbach [10]. For the reader’s convenience we outline its proof which can be also found in [2].

Proposition 3.6. (Stammbach) Let G be an R-torsion-free solvable group with the Hirsch rank h, where R is an integral domain of characteristic zero. Then hd_{R}(G) = h and Tor_{h}^{𝔽G}(𝔽, A) ≅ 𝔽, for some 𝔽G-module A isomorphic to 𝔽 as an 𝔽-module.

Proof. Let 1 = G_{0} ⊂ G_{1} ⊂ ... ⊂ G_{n} = G be a derived series of G. Let S_{i} = G_{i}/G_{i−1} and rk(S_{i}) = h_{i}. Set F_{i} = Tor_{h_{i}}^{𝔽S_{i}}(𝔽, 𝔽). From Proposition 3.5 it follows that F_{i} is additively isomorphic to 𝔽.

Denote by F_{i}^{op} the additive group of F_{i} with the inverse G-action. Let A = ⊗_{i=1}^{i=n}F_{i}^{op}. We use induction on n to prove Tor_{h}^{𝔽G}(𝔽, A) ≅ 𝔽. For n = 1 the result is clear. Suppose n ≥ 2, then G/G_{1} has derived length n − 1. Thus, by the Lyndon-Hochschild-Serre spectral sequence of the extension 1 → G_{1} → G → G/G_{1} → 1, we have

Tor_{h}^{𝔽G}(𝔽, A) ≅ Tor_{h−h_{1}}^{𝔽(G/G_{1})}(𝔽, Tor_{h_{1}}^{𝔽G_{1}}(𝔽, A)) ≅
Tor_{h−h_{1}}^{𝔽(G/G_{1})}(𝔽, Tor_{h_{1}}^{𝔽G_{1}}(𝔽, 𝔽) ⊗ A) ≅ Tor_{h−h_{1}}^{𝔽(G/G_{1})}(𝔽, ⊗_{i=2}^{i=n}F_{i}^{op}) ≅ 𝔽.

Now the assertion that hd_{R}(G) = h follows from the fact that h ≤ hd_{𝔽}(G) ≤ hd_{R}(G) ≤ h, when G is R-torsion-free. □

Theorem 3.7. Let G be a solvable group which acts freely and properly on an edimensional CW-complex E. Suppose there is an integer n ≤ e such that H_{i}(E, ℤ) = 0 for all i > n, and H_{n}(E, ℤ) ≅ ℤ, then h(G) ≤ e−n. If G is also without R-torsion, hd_{R}(G) ≤ e − n.

Proof. Suppose h(G) > e − n. Then, there exists a subgroup H of G which has finite Hirsch rank larger than e − n. Now by 3.6 it follows that hd_{ℚ}(H) = h(H). Therefore the homological dimension of H over ℚ is finite and larger than e − n, which contradicts Proposition 2.5. □

Example 3.8. Let F denote the Thompson group defined by the presentation

⟨x_{0}, x_{1}, x_{2}, ...|x_{i}^{−1}x_{n}x_{i} = x_{n+1} for all i < n and n ∈ ℕ⟩

This group is finitely presented for it also has the presentation

⟨x_{0}, x_{1}|x_{2}^{x_{0}} = x_{3}, x_{3}^{x_{1}} = x_{4}⟩

Brown and Goeghegan [4] showed that it is also of type FP_{∞}, i.e. it has a projective ℤG-resolution of ℤ, where each module is finitely generated. They also proved H_{n}(F, ℤ) = ℤ ⊕ ℤ for n ≥ 1 and H^{n}(F, ℤF) = 0 for all n. This may suggest that this group has periodic cohomology. But in fact, F cannot have periodic or jump cohomology over any ring, since it has an infinite rank abelian subgroup ℤ^{∞}. Namely, ⟨x_{0}x_{1}^{−1}, x_{2}x_{3}^{−1}, x_{4}x_{5}^{−1}, ...⟩ ⊆ F.

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This reinstates our belief that periodicity and jumps are rather special conditions on the cohomology of a discrete group, since countable groups containing the Thompson group are the only class of groups, known to us, not to be HF.

Corollary 3.9. If G is a solvable group that acts freely and properly on an (m + n)- dimensional complex homotopy equivalent to a closed, orientable manifold of dimension n, then the Hirsch rank h(G) ≤ m.

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Corollary 3.10. If G has a non-countable, torsion-free, solvable subgroup, then G cannot act freely and properly on a finite dimensional complex homotopy equivalent to a closed, orientable manifold.

Proof. Let H be a non-countable, torsion-free, solvable subgroup of G. By [2], Lemma 7.9, p. 100, H has infinite Hirsch rank. Therefore it cannot act freely and properly on a finite dimensional complex homotopy equivalent to a closed, orientable manifold. □

4. Actions of Height One

These next results apply only to countable torsion-free groups. We find an obstruction for a group to be free in a specific setting. It would be interesting to know whether the following proposition holds for not necessarily countable groups and what implications there are when the obstruction does not vanish.

Proposition 4.1. Let G be a finitely generated torsion-free group that acts freely and properly on an n-dimensional CW-complex X such that the G-invariant submodule (H_{n−1}(X, R))^{G} ≠ 0. Then G is a free group if and only if Ext_{RG}^{2}(R, H_{n}(X, R)) = 0.

Proof. If G is a free group then cd_{R}(G) ≤ 1, so Ext_{RG}^{2}(R, H_{n}(X, R)) must be trivial.

Conversely, suppose Ext_{RG}^{2}(R, H_{n}(X, R)) = 0. Let (C_{∗}(X), d_{∗}) be the chain complex of X, where C_{i}(X) is the free R-module generated by the i-cells of X. We proceed by induction on the number of generators of G with the case of G having exactly one generator understood.

There exists an exact sequence of ℤG-modules

0 → d_{n}(C_{n}(X)) →ker(d_{n−1}) → H_{n−1}(X, R) → 0

and an associated long exact sequence

Ext_{RG}^{0}(R, ker(d_{n−1})) → Ext_{RG}^{0}(R, H_{n−1}(X, R)) → Ext_{RG}^{1}(R, d_{n}(C_{n}(X))) →

Ext_{RG}^{1}(R, ker(d_{n−1}))

The first term of the long exact sequence is the G-invariant submodule (ker(d_{n−1}))^{G}. This module is trivial, for it is a submodule of the G-invariant submodule (C_{n−1}(X))^{G}. But G is infinite and C_{n−1}(X) is a free RG-module, hence (C_{n−1}(X))^{G} = 0. Thus the second term, the G-invariant submodule (H_{n−1}(X, R))^{G}, of the long exact sequence injects into Ext_{RG}^{1}(R, d_{n}(C_{n}(X))). This shows that Ext_{RG}^{1}(R, d_{n}(C_{n}(X))) ≠ 0.

On the other hand there exists an exact sequence of RG-modules

0 → H_{n}(X, R) → C_{n}(X) → d_{n}(C_{n}(X)) → 0

The long exact sequence associated to this gives:

Ext_{RG}^{1}(R, H_{n}(X, R)) → Ext_{RG}^{1}(R, C_{n}(X)) → Ext_{RG}^{1}(R, d_{n}(C_{n}(X))) →
Ext_{RG}^{2}(R, H_{n}(X, R))

According to our assumption Ext_{RG}^{2}(R, H_{n}(X, R)) = 0, so the term Ext_{RG}^{1}(R, C_{n}(X)) surjects onto the module Ext_{RG}^{1}(R, d_{n}(C_{n}(X))), which is nontrivial. Therefore Ext_{RG}^{1}(R, C_{n}(X)) ≠ 0. Now since G is finitely generated and C_{n}(X) is a free RGmodule, we must have Ext_{RG}^{1}(R, RG) ≠ 0. So by [5] G must split as a nontrivial free product of groups.

Suppose then G = H ∗ K where H, K ≠ G. For any RG-module M,

Ext_{RG}^{2}(R, M) = Ext_{RH}^{2}(R, M)⊕ Ext_{RK}^{2}(R, M)

It then follows that,

Ext_{RH}^{2}(R, H_{n}(X, R)) = Ext_{RK}^{2}(R, H_{n}(X, R)) = 0.

Also by Grushko’s Theorem [5] there exist a fewer number of generators for each of the groups H and K than the number of generators of G. So we can apply induction on these subgroups to show that they are free.

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Theorem 4.2. If G is a torsion-free countable group that acts freely and properly on an n-dimensional complex homotopy equivalent to a closed, orientable manifold of dimension n − 1, then G must be free.

Proof. First, let us assume G is finitely generated. If the induced action of G on H_{n−1}(X) = ℤ is nontrivial then G must contain an index two subgroup H which acts trivially on H_{n−1}(X). Thus, there always exists such H ⊆ G acting trivially, with [G : H] ≤ 2. Also, since H_{n}(X) = 0, H^{2}(H, H_{n}(X)) = 0. By Proposition 4.1 it follows that H is free. Since H is a finite index subgroup, G must be free.

In general, since G is a countable group, it is a countable union of an ascending chain of finitely generated subgroups. By the Theorem of Berstein, [6], we have cd(G) ≤ 2. Using Proposition 2.5, it follows that cd(G) ≤ 1. Therefore G is free.

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[3] Brown, K., Cohomology of Groups, Graduate Text in Math. 87, Springer-Verlag, New York 1982.

[4] Brown, K., and Geoghegan, R., An infinite-dimensional torsion-free FP_{∞} group, Invent. Math. 77 (1984) 367–381.

[5] Cohen, D., Groups of Cohomological Dimension One, Lecture Notes in Mathematics, 245, Berlin Heidelberg New-York, Springer-Verlag 1972.

[6] Gruenberg, K., Cohomological Topics in Group Theory, Lecture Notes in Mathematics, 143, Berlin Heidelberg New-York, Springer-Verlag 1970.

[7] Kropholler, P., On groups of type FP_{∞}, J. Pure. Appl. Algebra 90(1993), 55–67.

[8] Mislin, G., and Talelli, O., On groups which act freely and properly on finite dimensional homotopy spheres GAMA Proc. LMS 275 (2000).

[9] Serre, J.-P., C ohomologie des groupes discretes. Annals of Math. Studies 70 , Princeton (1971).

[10] Stammbach, U., On the weak homological dimension of the group algebra of solvable groups. J. London Math. Soc. (2) 2 (1970), 567–570.

[11] Talelli, O., Periodicity in Group Cohomology and Complete Resolutions, to appear in Bull. of LMS. 37 (2005), 547–554.

Department of Mathematics, Indiana University, Bloomington IN 47405, USA

E-mail address: nanpetro@indiana.edu