CLASSIFYING SPACES FOR PROPER

ACTIONS OF LOCALLY-FINITE GROUPS

Warren Dicks, Peter H. Kropholler, Ian J. Leary and Simon Thomas

February 22, 2002

For each finite ordinal n, and each locally-finite group G of cardinality ℵ_{n}, we construct an (n + 1)-dimensional, contractible CW-complex on which G acts with finite stabilizers. We use the complex to obtain information about cohomology with induced coefficients. Our techniques also give information about the location of some large free abelian groups in the hierarchy H 𝔉.

Throughout, let G be a group, and let A be a ℤ-module with trivial G-action. We let AG denote the induced ℤG-module ℤG ⊗_{ℤ} A. Ring-actions on modules and group-actions on sets are tacitly understood to be on the left, if not specified otherwise.

1. Holt’s Conjectures

The purpose of this section is to describe our main algebraic results, Theorems 3.10 and 5.4, and place them in the context of prior work.

1.1 Notation. Let rank(G) denote the smallest cardinal κ such that there exists some set of generators of G of cardinality κ.

If G is not finitely generated, then rank(G) = |G|, and we define ℵ-rank(G) to be the ordinal α such that rank(G) = ℵ_{α}; if G is finitely generated, then rank(G) < |G|, and we set ℵ-rank(G) = −1.

Recall that, for each ordinal α, ω_{α} denotes the least ordinal of cardinality ℵ_{α}. We find it convenient to set ℵ_{−1} = 1. Throughout this section, let n ∈ ℕ (= ω_{0}). □

D. F. Holt proposed the following description of the cohomology with induced coefficients, for locally-finite groups.

1.2 Conjecture (Holt [6]). If G is locally finite, then

| H^{n}(G, AG)| = |A|^{ℵ_{n−1}} if n = ℵ-rank(G) + 1,
H^{n}(G, AG) = 0  if n ≠ ℵ-rank(G) + 1.

Commentary. In Examples 3.3 we recall that, for any group G,

(1.3)  H^{n}(G, AG) = \{ \begin{matrix}A & if G is finite and n = 0, \\ 0 & if G is finite or n = 0, but not both. \end{matrix}

1991 Mathematics Subject Classification. Primary 20F50; Secondary 20J06, 20K20, 57M07.

Thus the conjecture really concerns the cases where n ≥ 1 and G is infinite, and the notation has been artificially contrived to embrace the trivial marginal cases.

For any infinite group G, the set of cocycles for G with coefficients in AG is of cardinality |A|^{|G|}, so 1 ≤ | H^{n}(G, AG)| ≤ |A|^{|G|}. If G is infinite and locally finite, then the conjecture implies that only the extreme values can be achieved. □

We now briefly state the cases which are known, including those obtained in this paper.

1.4 Notation. We say that G has the finite extension property for proper subgroups if each proper subgroup of G is a proper subgroup of finite index in some subgroup of G. For example, abelian torsion groups have this property.

Let us say that A is o(G)-inverting if, for every finite subgroup H of G, multiplication by |H| gives an automorphism of A; equivalently, for each g ∈ G whose order o(g) is finite, multiplication by o(g) gives an automorphism of A.

If R is a ring (associative, with 1), then R is o(G)-inverting, as ℤ-module, if and only if the order of each finite subgroup of G is a unit in R. If R is not o(G)-in-verting, then it is easy to show that cd_{R} G, the cohomological dimension of G with respect to R, is ∞, a value which we shall think of as ω_{0} = ℵ_{0}. □

1.5 Known cases of Conjecture 1.2. Let G be a locally-finite group. (1) H^{n}(G, AG) = 0 if n > ℵ-rank(G) + 1.

(2) If G has the finite extension property for proper subgroups, then H^{n}(G, AG) = 0 if n ≠ ℵ-rank(G) + 1.

(3) For n ∈ {0, 1}, H^{n}(G, AG) = 0 if n ≠ ℵ-rank(G) + 1.

(4) For n ∈ {0, 1, 2}, | H^{n}(G, AG)| = |A|^{ℵ_{n−1}} if n = ℵ-rank(G) + 1.

(5) It is consistent with ZFC that | H^{n}(G, AG)| ≥ 2^{ℵ_{n−1}} if n = ℵ-rank(G) + 1 and A is nonzero. Hence, it is consistent with ZFC that | H^{n}(G, AG)| = |A|^{ℵ_{n−1}} if n = ℵ-rank(G) + 1 and |A| ≤ ℵ_{n−1}.

Commentary. (1), the “easy” part of Conjecture 1.2, is proved in Theorem 3.10. It was proved in [11] for the case where A is o(G)-inverting, and, before that, in [4], [5] for the case where A is o(G)-inverting and torsion.

(2) was proved by Holt [5]. We give another proof of the abelian case in Corollary 6.10.

(3). By (1.3), this holds for n = 0. It was proved by Holt [6] for n = 1; see Theorem 6.4.

(4). By (1.3), this holds for n = 0. It is well known for n = 1; see Theorem 4.5. In Theorem 5.4, we prove it for n = 2; Holt [6] had previously shown this was consistent with ZFC, see [14, Section 1].

(5). Suppose that n = ℵ-rank(G) + 1 and that A is nonzero.

We shall now see that it is consistent with ZFC that | H^{n}(G, AG)| ≥ 2^{ℵ_{n−1}}. For each prime p, we write ℤ(p^{∞}) := lim_{m→∞} ℤ/p^{m}ℤ, where, for m ∈ ℕ, the map

ℤ/p^{m}ℤ → ℤ/p^{m+1}ℤ is given by multiplication by p.

We claim that there exists a ℤ-module k, and a ℤ-submodule A′ of A, such that the quotient A/A′ is isomorphic to k, and either k = ℚ, or there exists a prime p such that k = ℤ/pℤ or k = ℤ(p^{∞}).

Consider first the case where A is not divisible, so there exists a prime p such that A/pA is nonzero. But A/pA is a direct sum of ℤ-submodules each of which is isomorphic to ℤ/pℤ. Hence A/pA projects onto any such summand.

If A is divisible, then A is a direct sum of ℤ-submodules each of which is isomorphic to ℚ or to ℤ(p^{∞}) for some prime p; see, for example, [3, Theorem IV.23.1]. Hence A projects onto any such summand.

In all cases, we can find A′, k as claimed.

Now there is a long exact sequence in cohomology which contains the subsequence

H^{n}(G, AG) → H^{n}(G, kG) → H^{n+1}(G, A′G).

By (1), H^{n+1}(G, A′G) = 0, so | H^{n}(G, AG)| ≥ | H^{n}(G, kG)|.

Thus, for the first part, it remains to show that it is consistent with ZFC that | H^{n}(G, kG)| ≥ 2^{ℵ_{n−1}}.

It is proved in [11] that, if k is an o(G)-inverting prime field, then it is consistent with ZFC that | H^{n}(G, kG)| ≥ 2^{ℵ_{n−1}}. However on carefully reading that proof, one sees that all applications of the o(G)-inverting hypothesis can be replaced with applications of (1), so, in fact, it is proved that it is consistent with ZFC that if k is a prime field then | H^{n}(G, kG)| ≥ 2^{ℵ_{n−1}}.

It remains to consider the case where k = ℤ(p^{∞}) for some prime p. Again, it is not difficult to show that the argument in [11] can be further modified to cover this case by interpreting dim k^{m} := m for all m ∈ ℕ. If M is a finitely generated (free) ℤ-submodule of ℤG, and kM denotes the image of the natural map k ⊗_{ℤ} M → k ⊗_{ℤ} ℤG = kG, then one can show dim kM = rank M, since k is divisible. Moreover, if M′ is a ℤ-submodule of M, then dim kM′ ≤ dim kM, and, if equality holds, then kM′ = kM, since k is divisible. Using these observations, one can verify that the argument in [11] applies with k = ℤ(p^{∞}).

It follows that, in all cases, it is consistent with ZFC that | H^{n}(G, kG)| ≥ 2^{ℵ_{n−1}}. Now suppose that |A| ≤ ℵ_{n−1}. Hence |A|^{ℵ_{n−1}} = 2^{ℵ_{n−1}}; see [7, p.49] for the case

where n ≥ 1. Thus, it is consistent with ZFC that | H^{n}(G, AG)| ≥ |A|^{ℵ_{n−1}}.

We previously observed that | H^{n}(G, AG)| ≤ |A|^{ℵ_{n−1}}, so it is consistent with ZFC that | H^{n}(G, AG)| = |A|^{ℵ_{n−1}}.

This proves (5). It had previously been proved by Holt [5] in the case where G has the finite extension property for proper subgroups; see [14, Section 1]. □

We wish to refine part of Conjecture 1.2.

1.6 Conjecture. If G is locally finite, and n = ℵ-rank(G) + 1, then H^{n}(G, AG) contains a ℤ-submodule isomorphic to A^{ℵ_{n−1}}, and hence | H^{n}(G, AG)| = |A|^{ℵ_{n−1}}.

Commentary. By (1.3), this holds for n = 0. It is probably well known for n = 1; see Theorem 4.5. In Theorem 5.4, we prove it for n = 2. □

Conjecture 1.2 was preceded by, and motivated by, an earlier proposal, concerning the cohomological dimension of locally-finite groups.

1.7 Conjecture. If G is a locally-finite group, and R is a nonzero, o(G)-inverting ring, then cd_{R} G = min{ℵ-rank(G) + 1, ∞}.

Commentary. Holt [4] proposed this conjecture with the additional hypothesis that R is a field of prime order, and, in [11], the additional hypothesis was weakened to R being commutative.

Notice that min{ℵ-rank(G) + 1, ∞} can be expressed as inf{n ∈ ℕ | ℵ_{n} > |G|}, where the infimum of the empty set is taken to be ∞.

The inequality cd_{R} G ≤ ℵ-rank(G) + 1 follows from a classic result of Auslander [1, Proposition 3]; see [12, Lemma 3.7] or Theorem 3.10 below.

Cohomological dimension cannot increase on passing to a subgroup, so we may assume that ℵ-rank(G) < ω_{0}, and let n = ℵ-rank(G) + 1. The conjecture now amounts to the claim that H^{n}(G, M) ≠ 0 for some RG-module M. Notice that, on RG-modules, H^{n+1}(G, −) vanishes and (hence) H^{n}(G, −) is right exact; also, M is a quotient of some free RG-module. The conjecture is therefore equivalent to the claim that H^{n}(G, AG) ≠ 0 for some free R-module A. This claim is implied by the claim that H^{n}(G, AG) ≠ 0 for some vector space A over the prime subfield of some simple quotient ring of R. This means that we may assume that R is a prime field.

The foregoing claims are known to be consistent with ZFC [11], and have been proved in various cases. The case where G is abelian was proved by N. Chen; see [12, Corollary 7.6] or Corollary 6.10 below. Chen’s result has been extended in two directions. Osofsky [12, Corollary 7.5] settled the case where G is generated by finite groups whose pairwise products are subgroups; in particular, G has the finite extension property for proper subgroups. Holt [5] settled the case where G has the finite extension property for proper subgroups and R is a finite prime field. □

2. G-complexes

In this section we construct finite-dimensional contractible spaces with locallyfinite groups acting on them.

2.1 Definitions. A map f: X_{1} → X_{2} between CW-complexes is cellular if it carries the d-skeleton of X_{1} to the d-skeleton of X_{2} for all d ∈ ℕ.

A G-CW-complex, or G-complex for short, is a CW-complex X with a G-action such that each element of G acts continuously on X, permuting the open cells, and fixing only those cells which it fixes pointwise. It follows that G acts cellularly.

If X is a G-complex then, for each H ≤ G, the set X^{H}, consisting of points fixed by all of H, is a CW-subcomplex of X, and the set X/H consisting of the H-orbits is a quotient CW-complex.

Let 𝔰𝔲𝔟(G) denote the set of all subgroups of G. A subset 𝔛 of 𝔰𝔲𝔟(G) is a subgroup-closed G-family if each subgroup of each element of 𝔛 belongs to 𝔛 and, moreover, 𝔛 is closed under taking conjugates by elements of G.

If 𝔛 is a subgroup-closed G-family, then by a space of type E(G, 𝔛) we mean a G-complex X with the properties that, for each H ∈ 𝔛, X^{H} is contractible, and for each H ∈ 𝔰𝔲𝔟(G) − 𝔛, X^{H} is empty. In this event, X is also said to be a classifying space for G-actions with stabilizers in 𝔛.

If 𝔛 is a class of groups, and 𝔛 ∩ 𝔰𝔲𝔟(G) is a subgroup-closed G-family, then by a space of type E(G, 𝔛) we mean a space of type E(G, 𝔛 ∩ 𝔰𝔲𝔟(G)).

We let 𝔉 denote the class of finite groups. Notice that 𝔉 ∩ 𝔰𝔲𝔟(G) is a subgroupclosed G-family. A space of type E(G, 𝔉) is called an \underline{E}G and a classifying space for proper G-actions. □

The following is well known.

2.2 Proposition. If 𝔛 is a subgroup-closed G-family, then there exists a space of type E(G, 𝔛), and any G-map between two spaces of type E(G, 𝔛) is a G-homotopy equivalence.

Proof. The first part can be seen by Milnor’s construction. Thus, let Δ be any G-set such that 𝔛 is precisely the set of subgroups of G which fix at least one point of Δ. Let X = Δ ∗ Δ ∗ Δ ∗ ⋯, the union of iterated joins of Δ. Then X is a space of type E(G, 𝔛).

For the second part, see, for example, [15, Proposition II.2.7]. □

2.3 Corollary. If 𝔛_{1} ⊆ 𝔛_{2} are subgroup-closed G-families, and X_{1} (resp. X_{2}) is a space of type E(G, 𝔛_{1}) (resp. E(G, 𝔛_{2})), then there exists a cellular G-map X_{1} → X_{2}.

Proof. The join X_{1} ∗ X_{2} is a space of type E(G, 𝔛_{2}), and the inclusions

ι_{1}: X_{1} → X_{1} ∗ X_{2}, ι_{2}: X_{2} → X_{1} ∗ X_{2}

are G-maps. By Proposition 2.2, ι_{2} is a G-homotopy equivalence, and the homotopy inverse X_{1} ∗ X_{2} → X_{2} composed with ι_{1} gives a G-map X_{1} → X_{2}. This is then G-homotopic to a cellular G-map X_{1} → X_{2}; see, for example, [15, Theorem II.2.1]. □

One could give a dual proof, using the projection maps from the Cartesian product X_{1} × X_{2}, which is a space of type E(G, 𝔛_{1}).

The following is a topological analogue of a classic result of Auslander [1, Proposition 3].

2.4 Theorem. Let β be a limit ordinal, let (G_{α} | α ≤ β) be a continuous chain of subgroups of G, and let (𝔛_{α} | α ≤ β) be a continuous chain of subsets of 𝔰𝔲𝔟(G) such that, for each α ≤ β, 𝔛_{α} is a subgroup-closed G_{α}-family.

Let n ∈ ℕ, and suppose that, for each α < β, there exists an n-dimensional space Y_{α} of type E(G_{α}, 𝔛_{α}). Then there exists an (n + 1)-dimensional space of type E(G_{β}, 𝔛_{β}).

Proof. For each α < β, 𝔛_{α} ⊆ 𝔛_{α+1} ∩ 𝔰𝔲𝔟(G_{α}) are subgroup-closed G_{α}-families. Also, Y_{α} is a space of type E(G_{α}, 𝔛_{α}), and Y_{α+1} can be viewed as a space of type E(G_{α}, 𝔛_{α+1} ∩ 𝔰𝔲𝔟(G_{α})). By Corollary 2.3, there exists a cellular G_{α}-map Y_{α} → Y_{α+1}, and hence a cellular G_{α+1}-map f_{α}: G_{α+1} ×_{G_{α}} Y_{α} → Y_{α+1}. Let M_{α} denote the mapping cylinder of f_{α}. Since f_{α} is cellular, M_{α} has the structure of a CW-complex, and is a space of type E(G_{α+1}, 𝔛_{α+1}). Notice that dim M_{α} = n + 1, since dim Y_{α} = dim Y_{α+1} = n.

We recursively construct a continuous chain (X_{α} | α < β) where X_{α} is a space of type E(G_{α}, 𝔛_{α}), and, for α ≥ 1, dim X_{α} = n + 1.

We take X_{0} = Y_{0}, and at limit ordinals we take directed unions.

Suppose α < β and that X_{α} has been constructed.

By Proposition 2.2, since X_{α} and Y_{α} are of type E(G_{α}, 𝔛_{α}) there exists a cellular G_{α}-map Y_{α} → X_{α}, and hence a cellular G_{α+1}-map G_{α+1} ×_{G_{α}} Y_{α} → G_{α+1} ×_{G_{α}} X_{α}. Take X_{α+1} to be the identification space, or pushout,

G_{α+1} ×_{G_{α}} Y_{α} ⟶ M_{α}
↓  ↓
G_{α+1} ×_{G_{α}} X_{α} ⟶ X_{α+1}.

Notice that dim X_{α+1} = n + 1, since dim Y_{α} = n, dim M_{α} = n + 1, and dim X_{α} ≤ n + 1. It is not difficult to check that X_{α+1} is of type E(G_{α+1}, 𝔛_{α+1}).

This completes the proof. □

2.5 Remark. For n = 0 and β = ω_{0}, the construction in the above proof gives the Bass-Serre tree of the graph of groups corresponding to the countable ascending

chain (G_{α} | α < ω_{0}). Here, 𝔛_{ω_{0}} = ⋃_{α<ω_{0}} 𝔰𝔲𝔟(G_{α}). □

2.6 Theorem. If n ∈ ℕ, and G is a locally-finite group with ℵ-rank(G) < n, then there exists an n-dimensional \underline{E}G.

Proof. We argue by induction on n.

If n = 0, then G is finite. Here a single point with trivial G-action is a 0-dimensional \underline{E}G.

Thus we may assume that n ≥ 1, and that the result holds for smaller n.

We can choose a continuous chain (G_{α} | α ≤ ω_{n}) of subgroups of G such that G_{ω_{n}} = G, and, for each α < ω_{n}, ℵ-rank(G_{α}) < n − 1, so, by the induction hypothesis, there exists an (n − 1)-dimensional \underline{E}G_{α}. Thus, by Theorem 2.4, there exists an n-dimensional \underline{E}G.

This completes the proof. □

2.7 Remarks. The foregoing construction applies in greater generality.

Suppose that, for every group H, there is specified a subgroup-closed H-family 𝔜(H) satisfying the following three conditions:

Any group isomorphism H_{1} → H_{2} induces a bijection 𝔜(H_{1}) → 𝔜(H_{2}).

If H_{1} ≤ H_{2}, then 𝔜(H_{1}) ⊆ 𝔜(H_{2}).

If H is the union of a well-ordered chain of subgroups H_{α}, then 𝔜(H) is the union of the 𝔜(H_{α}).

Let 𝔊_{0} denote the class consisting of those groups H such that 𝔜(H) = 𝔰𝔲𝔟(H). For n < ω_{0}, recursively define 𝔊_{n+1} to be the class consisting of those groups which can be expressed as the union of a well-ordered chain of subgroups which lie in 𝔊_{n}.

The above argument then shows that if G ∈ 𝔊_{n}, then there exists an n-dimen-sional E(G, 𝔜(G)).

For any n > 0 and any G ∈ 𝔊_{n}, it can be arranged that all of the spaces involved in the construction of E(G, 𝔜(G)) have distinguished contractible subcomplexes which are transversals for the group actions. In particular, the quotient complex E(G, 𝔜(G))/G is contractible, although we will not use this information. □

We record one example.

2.8 Theorem. If n ∈ ℕ, and ℵ-rank(G) < n, and 𝔛 is the set of all subgroups of all finitely generated subgroups of G, then there exists an n-dimensional E(G, 𝔛).

Proof. For each group H, let 𝔜(H) be the set consisting of the subgroups of the finitely generated subgroups of H. It is easy to see that 𝔜(−) respects isomorphisms, inclusions and well-ordered unions.

It can be shown, by induction on n, that G ∈ 𝔊_{n}, in the notation of the previous remark, so, by that remark, there exists an n-dimensional E(G, 𝔜(G)). □

2.9 Example. In the foregoing theorem, if G is abelian (or locally finite), then 𝔛 is the set of finitely generated subgroups of G. □

3. Eventual vanishing of cohomology with induced coefficients

In this section, we recall how H^{∗}(G, −) can be computed using an \underline{E}G, and apply the method in the case where G is locally finite.

3.1 Definitions. Let M be a ℤG-module.

We say that M is G-acyclic if H^{n}(G, M) = 0 for all n ≥ 1.

Any ℤG-summand of a G-acyclic ℤG-module is again G-acyclic.

If ⋯ → P_{1} → P_{0} → ℤ → 0 is a ℤG-projective resolution of ℤ, then it is easy to see that M is G-acyclic if and only if the sequence

0 → Hom_{ℤG}(ℤ, M) → Hom_{ℤG}(P_{0}, M) → Hom_{ℤG}(P_{1}, M) → ⋯

is exact. When this holds, we say that Hom_{ℤG}(−, M) carries augmented ℤG-pro-jective resolutions of ℤ to exact sequences.

For any set Δ, we let A[[Δ]] denote the set of all functions from Δ to A, and such a function x ↦ a_{x} will be written as a formal sum ∑_{x∈Δ} a_{x}.x, and its support is

supp(∑_{x∈Δ} a_{x}.x): = {x ∈ Δ | a_{x} ≠ 0}.

The ℤ-module structure of A[[Δ]] is defined in the obvious way. We let A[Δ], or AΔ denote the ℤ-submodule of A[[Δ]] consisting of the elements with finite support. If Δ is a G-set, we view A[[Δ]] as a ℤG-module, with

g(∑_{x∈Δ} a_{x}.x) = ∑_{x∈Δ} a_{x}.gx = ∑_{x∈Δ} a_{g^{−1}x}.x.

(Notice that the action of G on the set of all functions from Δ to A is traditionally, and more naturally, on the right, with

(x ↦ a_{x})g = (x ↦ a_{gx}),

and our action of g on the left corresponds to the usual action of g^{−1} on the right. For our purposes, it is convenient to have actions on the left, wherever possible.) Then AΔ is a ℤG-submodule of A[[Δ]]. In the case where Δ = G, this notation is consistent with the notation for the induced ℤG-module AG. Here A[[G]] is called a coinduced ℤG-module.

There is a natural bijection

(3.2)  Hom_{ℤ}(M, A) → Hom_{ℤG}(M, A[[G]]), ψ ↦ (m ↦ ∑_{x∈G} ψ(x^{−1}m).x).

Since Hom_{ℤ}(−, A) carries ℤ-split exact sequences of ℤG-modules to ℤG-split exact sequences of ℤG-modules, we see that co-induced ℤG-modules are G-acyclic. □

3.3 Examples. Let G be a finite group.

Here, AG = A[[G]], so induced ℤG-modules are co-induced, and hence G-acyclic. Suppose that M is an o(G)-inverting ℤG-module. Then the multiplication map

M[[G]] → M, ∑_{g∈G} m_{g}.g ↦ ∑_{g∈G} gm_{g},

is ℤG-split with right inverse m ↦ \frac{1}{|G|} ∑_{g∈G} g^{−1}m.g. Here, M is a ℤG-summand of

an induced ℤG-module, so M is G-acyclic. □

In the following, G acts on tensor products over ℤ via the diagonal action.

3.4 Lemma. Let M be a ℤG-module.

(1) The functor Hom_{ℤG}(−⊗_{ℤ}ℤG, M) carries ℤ-split exact sequences of ℤG-mo-dules to exact sequences.

(2) Let H be a subgroup of G. If M is H-acyclic, then the functor

Hom_{ℤG}(ℤ[G/H] ⊗_{ℤ} −, M)

carries augmented ℤG-projective resolutions of ℤ to exact sequences.

Proof. (2). Let L be a ℤG-module. There is a natural identification of ℤG-modules,

ℤ[G/H] ⊗_{ℤ} L = ℤG ⊗_{ℤH} L,

with gH ⊗ ℓ corresponding to g ⊗ g^{−1}ℓ.

It follows that we can identify

Hom_{ℤG}(ℤ[G/H] ⊗_{ℤ} −, M) = Hom_{ℤG}(ℤG ⊗_{ℤH} (−, M) = Hom_{ℤH}(−, M)

as functors on ℤG-modules. Since M is H-acyclic, this functor carries augmented ℤG-projective resolutions of ℤ to exact sequences.

(1) is proved similarly. □

3.5 Notation. Let X be a G-complex.

We shall treat X as the G-set whose elements are the open cells of X. The cellular chain complex of X is then the permutation module ℤX, with the structure of a differential graded ℤG-module, with differential ∂ of degree −1. Here the grading is that determined by the dimensions of the cells, so the nth component C_{n}(ℤX) has as ℤ-basis the cells of dimension n.

We let η: X × X → ℤ denote the function such that ∂x = ∑_{y∈X} η(x, y).y for each

x ∈ X. Thus, if x is an n-cell, then η(x, y) = 0 unless y is one of the finitely many (n − 1)-cells incident to x. □

The following is a degenerate case of the equivariant cohomology spectral sequence; see, for example, [2, VII.7.10(7.10)].

3.6 Theorem. Let X be an acyclic G-complex. If M is a ℤG-module which is G_{x}-acyclic for each x ∈ X, then H^{∗}(Hom_{ℤG}(ℤX, M)) ≃ H^{∗}(G, M), as graded abelian groups.

Recall that Hom_{ℤG}(ℤX, M) denotes the differential graded abelian group with nth component C^{n}(Hom_{ℤG}(ℤX, M)) = Hom_{ℤG}(C_{n}(ℤX), M).

Proof. The homology of (ℤX, ∂) is ℤ, concentrated in degree zero.

We choose a free ℤG-resolution of ℤ, and write it as (ℤY, ∂) for some G-free G-set Y; for example, we could take Y to be an E G, and ℤY its cellular chain complex. Then Hom_{ℤG}(ℤY, M) is an additive abelian differential graded group, and its cohomology is H^{∗}(G, M).

We consider the double complex ℤX ⊗_{ℤ} ℤY with diagonal G-action, and the double complex Hom_{ℤG}(ℤX ⊗_{ℤ} ℤY, M). We get a fourth-quadrant commuting diagram which can be schematically represented as

(3.7)  0 ; ↓ ; Hom_{ℤG}(ℤX, M) ; ↓ ; 0 ⟶ Hom_{ℤG}(ℤY, M) ⟶ Hom_{ℤG}(ℤX ⊗_{ℤ} ℤY, M).

To show that the cohomology group of the outer row, Hom_{ℤG}(ℤX, M), is isomorphic to the cohomology group of the outer column, Hom_{ℤG}(ℤY, M), it suffices to show that the remaining, or inner, rows and columns of (3.7) are exact. Each inner column is exact because Hom_{ℤG}(ℤX ⊗_{ℤ} −, M) is exact on augmented projective ℤG-resolutions of ℤ, by Lemma 3.4(2). Similarly, each inner row is exact because Hom_{ℤG}(− ⊗_{ℤ} ℤY, M) is exact on ℤ-split exact sequences of ℤG-modules, by Lemma 3.4(1). □

3.8 Corollary. Let M be a ℤG-module, and let X be a finite-dimensional acyclic G-complex. If M is G_{x}-acyclic for each x ∈ X, then H^{n}(G, M) = 0 for all n > dim X. □

We record the case of finite stabilizers.

3.9 Corollary. Let M be a ℤG-module, and suppose that M is H-acyclic for each finite subgroup H of G; for example, this holds if M = AG, or if M is o(G)-inverting. Let X be an acyclic G-complex with finite stabilizers; for example, this holds if X is an \underline{E}G. Then H^{n}(G, M) = 0 for all n > dim X. □

Here we can apply Theorem 2.6.

3.10 Theorem. Let G be a locally-finite group, and M a ℤG-module which is H-acyclic for each finite subgroup H of G; for example, this holds if M = AG, or if M is o(G)-inverting. Then H^{n}(G, M) = 0 for all n > ℵ-rank(G) + 1. □

3.11 Remark. Theorem 3.10 can also be proved using the argument of the first paragraph of [11, Section 1]. □

4. Locally-finite groups of cardinality ℵ_{0}

In this section, we recall how H^{∗}(G, AG) can be computed using an \underline{E}G, and apply the method in the one-dimensional case.

4.1 Definitions. Let X be an \underline{E}G, or, more generally, any acyclic G-complex in which all cell stabilizers are finite, and let Notation 3.5 apply.

We have natural identifications

(4.2)  A[[X]] = A^{X} = Hom_{ℤ}(ℤX, A).

For simplicity, let us suppose that X is finite dimensional.

Then A[[X]] has the structure of a differential graded ℤG-module, in which the differential ∂^{∗} has degree +1, and is given by

∂^{∗}(∑_{x∈X} a_{x}.x) = ∑_{x∈X}(∑_{y∈X} η(x, y)a_{y}).x.

The cohomology of (A[[X]], ∂^{∗}) is A concentrated in degree zero.

Let

A_{G}[[X]] := {∑_{x∈X} a_{x}.x ∈ A[[X]] | {g ∈ G | a_{gx} ≠ 0} is finite, for all x ∈ X}.

Since G-stabilizers are finite, we see that A_{G}[[X]] consists of all functions from X to A with finite support in each G-orbit.

It is straightforward to check that A_{G}[[X]] is a differential graded ℤG-submodule of A[[X]].

We write C^{n}(A_{G}[[X]]), B^{n}(A_{G}[[X]]), and Z^{n}(A_{G}[[X]]) for the n-cochains, n-co-boundaries, and n-cocycles, respectively. □

Sometimes the notation Hom_{c}(ℤ[X], A) is used to denote A_{G}[[X]]; see, for example, [2, Lemma VIII.7.4].

The following is a variation on the usual “compact supports” cohomology; see, for example, [2, Proposition VIII.7.5]. It is particularly useful in the study of ends of groups.

4.3 Theorem. If X is a finite-dimensional acyclic G-complex with finite stabilizers, then there is a natural isomorphism H^{∗}(A_{G}[[X]]) ≃ H^{∗}(G, AG) of graded abelian groups.

Proof. There is a natural identification of Hom_{ℤG}(ℤX, A[[G]]) with Hom_{ℤ}(ℤX, A); see (3.2). There is also a natural identification of Hom_{ℤ}(ℤX, A) with A[[X]]; see (4.2). It is easy to show that under these identifications, Hom_{ℤG}(ℤX, AG) corresponds to A_{G}[[X]]. Hence H^{∗}(A_{G}[[X]]) ≃ H^{∗}(Hom_{ℤG}(ℤX, AG)). Finally, H^{∗}(Hom_{ℤG}(ℤX, AG)) ≃ H^{∗}(G, AG), by Theorem 3.6. □

There is a natural right G-action on H^{∗}(G, AG), arising from the ℤG-bimodule structure on AG. This agrees with the natural right G-action on A_{G}[[X]] which we have transformed into a left G-action.

Let us illustrate how Theorem 4.3 can be used to study H^{∗}(G, AG) when G is locally finite of cardinality ℵ_{0}. To do this we now construct a standardized \underline{E}G, as in Remark 2.5.

4.4 Definition. Let G be a locally-finite group of cardinality ℵ_{0}.

Index the elements of G with ω_{0}, so G = {h_{α} | α < ω_{0}}. For each β ≤ ω_{0}, let H_{β} := ⟨h_{α} | α < β⟩. Let (G_{α} | α ≤ ω_{0}) be the subsequence of (H_{α} | α ≤ ω_{0}) obtained by omitting each term which is equal to an earlier term.

Notice that G_{ω_{0}} = G and, for each α < ω_{0}, |G_{α}| < ℵ_{0}. Moreover, G_{0} = 1, and, for each α < ω_{0}, G_{α+1} = ⟨G_{α}, g_{α}⟩, where g_{α} = h_{α′} and α′ is the least ordinal such that h_{α′} ∉ G_{α}.

We define the line, denoted ℝ, to be the tree with vertices (v_{n} | n ∈ ℤ), (oriented) edges (e_{n} | n ∈ ℤ), and incidence relations ιe_{n} = v_{n}, τe_{n} = v_{n+1}, for all n ∈ ℤ.

We define the half-line, denoted ℝ^{+}, to be the subtree of ℝ with vertices (v_{α} | α < ω_{0}), and edges (e_{α} | α < ω_{0}).

There is a G-tree X with G-transversal ℝ^{+} such that the G-stabilizer of v_{α}, and of e_{α}, is G_{α}, for all α < ω_{0}. This completely specifies X. It is not difficult to see that X is a locally-finite tree with one end.

We denote the set of vertices (resp. edges) of X by V X (resp. EX). □

The following is fairly standard, but we do not know an explicit reference.

4.5 Theorem. If G is a locally-finite group of cardinality ℵ_{0}, then H^{∗}(G, AG) is concentrated in degree 1, H^{1}(G, AG) contains a ℤ-submodule isomorphic to A^{ℵ_{0}}, and | H^{1}(G, AG)| = |A|^{ℵ_{0}} = |A|^{|G|}.

Proof. By (1.3) and Theorem 3.10, H^{∗}(G, AG) is concentrated in degree 1, so it remains to study H^{1}(G, AG).

Let X be as in Definition 4.4.

By Theorem 4.3, H^{1}(G, AG) = H^{1}(A_{G}[[X]]).

Let Q denote {e_{α} | α < ω_{0}}, so we can view

A[[Q]] ⊆ A_{G}[[EX]] = C^{1}(A_{G}[[X]]) = Z^{1}(A_{G}[[X]]).

Let P be a partition of ω_{0} into ℵ_{0} subsets, each of cardinality ℵ_{0}. We denote the map ω_{0} → P by α ↦ [α].

Consider any ϕ ∈ A^{P}, [α] ↦ ϕ[α].

Define

ϕ^{†}: = ∑_{α<ω_{0}} ϕ[α].e_{α} ∈ A[[Q]] ⊆ Z^{1}(A_{G}[[X]]).

We claim that if ϕ^{†} ∈ B^{1}(A_{G}[[X]]) then ϕ = 0.

Suppose not, so there exists ψ ∈ C^{0}(A_{G}[[X]]) = A_{G}[[V X]] such that ∂^{∗}ψ = ϕ^{†}, and there exists p ∈ P such that ϕ(p) ≠ 0.

There exists μ < ω_{0} such that supp(ψ) ∩ Gv_{0} ⊆ G_{μ}v_{0}.

There exists α ∈ p such that α > μ. Notice ϕ[α] = ϕ(p) ≠ 0.

In ℤX, ∂(∑_{i=0}^{α} e_{i}) = v_{α+1} − v_{0}. Hence ∂((g_{α} − g_{α−1}) ∑_{i=0}^{α} e_{i}) = −(g_{α} − g_{α−1})v_{0},

since g_{α}, g_{α−1} ∈ G_{α+1} = G_{v_{α+1}}.

We can view ψ as an additive map ℤX → A, and apply it to the foregoing to

get ϕ^{†}((g_{α} − g_{α−1}) ∑_{i=0}^{α} e_{i}) = ψ(−(g_{α} − g_{α−1})v_{0}) = 0, since g_{α}, g_{α−1} ∉ G_{α−1} ⊇ G_{μ}.

Thus

0 = ϕ^{†}((g_{α} − g_{α−1}) ∑_{i=0}^{α} e_{i}) = ϕ^{†}(∑_{i=0}^{α} g_{α}e_{i}) − ϕ^{†}(∑_{i=0}^{α−1} g_{α−1}e_{i}) − ϕ^{†}(e_{α}) = 0 − 0 − ϕ(p).

Hence ϕ(p) = 0, which is a contradiction.

This proves that the composition

A^{P} → A[[Q]] → H^{1}(A_{G}[[X]]) = H^{1}(G, AG)

is injective. Since |P| = ℵ_{0}, A^{ℵ_{0}} embeds in H^{1}(G, AG). □

5. Locally-finite groups of cardinality ℵ_{1}

In this section we study H^{∗}(G, AG) when G is locally finite with ℵ-rank(G) = 1, topologizing and refining results of D. J. Holt.

We begin by constructing a standardized \underline{E}G.

5.1 Definitions. Let G be a locally-finite group of cardinality ℵ_{1}.

Let ω′_{1} denote the set of limit ordinals less than ω_{1}.

As in Definition 4.4, we start by indexing the elements of G, G = {h_{α} | α < ω_{1}}, set H_{β} := ⟨h_{α} | α < β⟩ for each β ≤ ω_{1}, and let (G_{α} | α ≤ ω_{1}) be the subsequence of (H_{α} | α ≤ ω_{1}) obtained by omitting each term which is either finite, or equal to an earlier term.

Notice (G_{α} | α ≤ ω_{1}) is a continuous chain of subgroups of G, G_{ω_{1}} = G, and, for each α < ω_{1}, |G_{α}| = ℵ_{0} and G_{α+1} = ⟨G_{α}, g_{α}⟩, where g_{α} = h_{α′}, and α′ is the least ordinal such that h_{α′} ∉ G_{α}.

For each subgroup H of G, we set H^{•} := G_{0} ∩ H.

We shall now construct a family (G_{α,n} | α < ω_{1}, n < ω_{0}) of finite subgroups of G such that the following hold:

(1) for each α < ω_{1}, (G_{α,n} | n < ω_{0}) is an increasing chain with union G_{α};

(2) for each α < ω_{1}, n < ω_{0}, G_{α+1,n} = ⟨G_{α,n}, g_{α}⟩;

(3) for each α ∈ ω′_{1}, n < ω_{0}, there is a distinguished element

g_{α,n} ∈ G_{α,n+1}^{•} − G_{α,n}^{•}(= G_{0} ∩ (G_{α,n+1} − G_{α,n}));

(4) for each α ∈ ω′_{1}, n < ω_{0}, G_{α+1,n} ∩ G_{α} = G_{α,n}.

We proceed as follows.

First, choose an arbitrary chain (G_{0,n} | n < ω_{0}) of finite subgroups with union G_{0}.

For α ∈ ω′, choose an arbitrary chain (G̃_{α,m} | m < ω_{0}) of finite subgroups with union G_{α}. For each m < ω_{0}, define G̃_{α+1,m} := ⟨G̃_{α,m}, g_{α}⟩. Choose any increasing function ω_{0} → ω_{0}, n ↦ m_{n}, such that the chain (G̃_{α+1,m_{n}}^{•} | n < ω_{0}) of finite subgroups of G_{0} with union G_{0} is strictly increasing. For n < ω_{0}, define

G_{α+1,n} := G̃_{α+1,m_{n}} and G_{α,n} := G_{α} ∩ G_{α+1,n}.

Notice that (4) holds, and that we may assume that (3) holds. Also G_{α,n} ⊇ G̃_{α,m_{n}}, so

G_{α+1,n} = G̃_{α+1,m_{n}} = ⟨G̃_{α,m_{n}}, g_{α}⟩ ⊆ ⟨G_{α,n}, g_{α}⟩ ⊆ G_{α+1,n}.

For α < ω_{1}, and n < ω_{0}, if G_{α,n} is defined, set G_{α+1,n} := ⟨G_{α,n}, g_{α}⟩; this completes the recursive definition of the family (G_{α,n} | α < ω_{1}, n < ω_{0}), and we see that (1) and (2) also hold.

For α < ω_{1}, let Y_{α} denote the G_{α}-tree of Definition 4.4 corresponding to the chain (G_{α,n} | n < ω_{0}).

We define the plane, denoted ℝ^{2}, to be the two-dimensional CW-complex with vertices (v_{m,n} | (m, n) ∈ ℤ^{2}), and edges (x_{m,n}, y_{m,n} | (m, n) ∈ ℤ^{2}), and faces (f_{m,n} | (m, n) ∈ ℤ^{2}), and incidence relations given by, for (m, n) ∈ ℤ^{2},

ιx_{m,n} = τx_{m−1,n} = ιy_{m,n} = τy_{m,n−1} = v_{m,n}

and f_{m,n} is attached along the path x_{m,n}, y_{m+1,n}, x_{m,n+1}^{−1}, y_{m,n}^{−1}.

We define the semi-infinite strip, denoted [0, 1] × ℝ^{+}, to be the subcomplex of ℝ^{2} with vertices (v_{0,n}, v_{1,n} | n < ω_{0}), edges (x_{0,n}, y_{0,n}, y_{1,n} | n < ω_{0}), and faces (f_{0,n} | n < ω_{0}).

Notice there are two distinguished subcomplexes of [0, 1]×ℝ^{+} which are isomorphic to ℝ^{+}, and will be denoted {0} × ℝ^{+} and {1} × ℝ^{+}.

Let α < ω_{1}. We construct a two-dimensional G_{α+1}-space M_{α}, for which [0, 1] × ℝ^{+} is a G_{α+1}-transversal, and the G_{α+1}-stabilizer of v_{0,n}, x_{0,n}, y_{0,n}, and f_{0,n} is G_{α,n}, while the G_{α+1}-stabilizer of v_{1,n} and y_{1,n} is G_{α+1,n}. This completely specifies M_{α}. Notice that M_{α} is the mapping cylinder of a G_{α+1}-map

G_{α+1} ×_{G_{α}} Y_{α} → Y_{α+1}.

If α ∈ ω′_{1}, then Y_{α} is a G_{α}-subtree of Y_{α+1}.

To construct X we glue together

(G ×_{G_{α}} M_{α} | α < ω_{1})

amalgamating

(G ×_{G_{α}} Y_{α} | α < ω_{1}),

as in the proof of Theorem 2.4. The image in X_{α+1} ⊆ X of f_{0,n} ∈ M_{α} will be denoted f_{α,n}.

We denote the set of vertices (resp. edges, resp. faces) of X by V X (resp. EX, resp. F X).

Let us recall the notion of a club (= closed unbounded subset). Thus, an ω_{1}-club is any subset S of ω_{1} such that the set of the least upper bounds of the nonempty subsets of S is precisely S ∪ {ω_{1}}. If S is an ω_{1}-club, then so is S ∩ ω′_{1}; recall that ω′_{1} denotes the set of limit ordinals in ω_{1}.

For ϕ ∈ A_{G}[[X]] and α ≤ ω_{1}, if supp(ϕ) ∩ GX_{α} ⊆ X_{α}, we say that ϕ respects α. It is straightforward to show that the set of ordinals in ω_{1} respected by ϕ is an ω_{1}-club. □

5.2 Lemma (Holt [6]). Let G be a periodic group, and let H, K be proper subgroups of G which generate G. Let X be the G-graph with vertex set the disjoint union of G/H and G/K, and edge set G/(H ∩ K), with g(H ∩ K) joining gH to gK, for each g ∈ G. Then X is connected, and deleting the two vertices H and K, and the one edge H ∩ K, leaves a connected space.

Proof. Collapsing all the edges of X leaves a transitive G-set, and one of the points is fixed by H and K. Since H and K generate G, this point is fixed by G, so forms a G-orbit. Thus we have only one point, so X is connected.

Let Y be the subgraph of X obtained by deleting the vertices H and K and all their incident edges. It suffices to show that Y is a connected graph. Since X is connected, it suffices to show that each X-neighbour of H is Y -connected to each X-neighbour of K. Thus, let h ∈ H − K, and k ∈ K − H; it suffices to show that the vertices hK and kH are Y -connected.

Let L = ⟨hk^{−1}⟩ = ⟨kh^{−1}⟩. Since G is periodic, L is finite.

We consider the action of L on X. Let m and n denote the orders of the L-orbits of the vertices H and K, respectively. By symmetry, we may assume that m ≥ n. Notice that L is not contained in K, so n ≥ 2.

Let g = kh^{−1}. In X, there is an edge H ∩ K joining H to K, and an edge k(H ∩ K) joining kH = gH to kK = K. Applying powers of g to these, we get a path in X with vertices

H, K, gH, gK, g^{2}H, g^{2}K, …, g^{n−1}H, g^{n−1}K.

By the definition of m and n, these 2n vertices are all distinct, so, on deleting the first two, we get a path in Y connecting gH = kH to g^{n−1}K = g^{−1}K = hK. □

5.3 Theorem (Holt [6]). If G is locally finite, and |G| = ℵ_{1}, then H^{1}(G, AG) = 0.

Proof. Let X be as in Definitions 5.1.

Consider any ϕ ∈ Z^{1}(A_{G}[[X]]). Thus supp(ϕ) is a collection of edges of X, with only finitely many in each G-orbit. A subset of X which meets (that is, has nonempty intersection with) supp(ϕ) is said to be broken by ϕ. Since ϕ is a 1-cocycle, we get 0 if we sum, in A, the ϕ-labels, with the appropriate signs, around any face, or along any closed path in the 1-skeleton, since X is simply-connected. Thus there is a well-defined ϕ-sum from any vertex to any other vertex.

Consider any α ∈ ω′_{1} such that ϕ respects α as in the last paragraph of Definitions 5.1.

From Definitions 5.1, there is a cellular G_{α+1}-map M_{α+1} → X, so ϕ induces an element ϕ_{α+1} ∈ A[[M_{α+1}]]. Since the G-stabilizers for X are finite, ϕ_{α+1} lies in A_{G_{α+1}}[[M_{α+1}]], Moreover, ϕ_{α+1} respects α in the obvious sense, since Y_{α} is mapped to X_{α}, by construction.

There exists n_{0} < ω_{0} such that supp(ϕ_{α+1}) ∩ G_{α+1}x_{0,0} ⊆ G_{α+1,n_{0}}x_{0,0}. This means that, for g ∈ G_{α+1}, if ϕ_{α+1} breaks gx_{0,0}, then the terminal vertex of gx_{0,0} lies in G_{α+1,n_{0}}v_{1,0}.

Now consider any n such that n_{0} < n < ω_{0}.

Let p_{n} denote the reduced open path in the tree {0} × ℝ^{+} from v_{0,0} to v_{0,n}, and e · p_{n} the open path obtained by concatenating e: = x_{0,0}^{−1} and p_{n}. We are interested in the G_{α+1,n}-graph Z generated by the closure \overline{e · p_{n}}.

Consider any g ∈ G_{α+1,n}.

If ϕ_{α+1} breaks gp_{n} then g ∈ G_{α}, since ϕ_{α+1} respects α, and hence g ∈ G_{α,n}, so gv_{0,n} = v_{0,n}. That is, if ϕ breaks gp_{n}, then the terminal vertex of gp_{n} is v_{0,n}.

We apply Lemma 5.2 to the graph Y′ obtained by taking H^{−} = G_{α+1,0} and K^{−} = G_{α,n_{0}}, so ⟨H^{−}, K^{−}⟩ = ⟨G_{α,n_{0}}, g_{α}⟩ = G_{α+1,n_{0}}. By Lemma 5.2, deleting g^{−1}K^{−} from Y′ leaves a connected space containing G_{α+1,n_{0}}/H^{−}, where we include the trivial case where g^{−1}K^{−} does not belong to Y′.

Let Y be the G_{α+1,n_{0}}-subspace of M_{α+1} generated by \overline{e · p_{n}}, and consider the G_{α+1,n_{0}}-map from Y to Y′ which assigns v_{1,0} to H^{−}, v_{0,n} to K^{−}, and e · p_{n} to H^{−} ∩ K^{−}. It follows that deleting g^{−1}K^{−}\overline{p_{n}} from Y leaves a connected space containing G_{α+1,n_{0}}v_{1,0}.

Suppose g ∉ G_{α+1,n_{0}}, so g ∈ G_{α+1,n} −G_{α+1,n_{0}}. Then supp(ϕ_{α+1})∩gY ⊆ K^{−}p_{n}. Hence, on deleting supp(ϕ_{α+1}) from the 1-skeleton of M_{α+1}, one of the resulting components contains gG_{α+1,n_{0}}v_{1,0}.

Next, we apply Lemma 5.2 to the graph Z′ obtained by taking H = G_{α+1,n_{0}} and K = G_{α,n}, so ⟨H, K⟩ = G_{α+1,n}. We conclude that

Z′ − ({H} ∪ {K} ∪ (H ∪ K)/(H ∩ K))

is a connected graph.

Let Z be the G_{α+1,n}-subspace of M_{α+1} generated by \overline{e · p_{n}}, and consider the map of G_{α+1,n}-spaces from Z to Z′ which assigns v_{1,0} to H, v_{0,n} to K, and e · p_{n} to H ∩ K. There is induced a surjective map

Z − ({v_{1,0}} ∪ {v_{0,n}} ∪ (H ∪ K)(e · p_{n})) → Z′ − ({H} ∪ {K} ∪ (H ∪ K)/(H ∩ K)).

Notice that ϕ_{α+1} breaks only edges of Z which lie in He∪Kp_{n}, so there is a map from Z −({v_{1,0}}∪{v_{1,n}}∪(H ∪K)(e·p_{n})) to the set of components of the 1-skeleton of M_{α+1} − supp(ϕ_{α+1}). Moreover, we have seen that each subset gHv_{1,0} maps to a component of the 1-skeleton of M_{α+1} − supp(ϕ_{α+1}). Thus the map factors through Z′ − ({H} ∪ {K} ∪ (H ∪ K)/(H ∩ K)), which is connected, so maps to a single component. Hence some component X′ of the 1-skeleton of M_{α+1} − supp(ϕ_{α+1}) contains (⟨H, K⟩ − H)v_{1,0}, that is, (G_{α+1,n} − G_{α+1,n_{0}})v_{1,0}.

Since n > n_{0} was arbitrary, all of (G_{α+1} −G_{α+1,n_{0}})v_{1,0} is contained in X′. Thus, for any path between any two elements of (G_{α+1} −G_{α+1,n_{0}})v_{1,0}, the ϕ_{α+1}-sum, and hence the ϕ-sum, is 0.

Let ψ_{α+1} ∈ C^{0}(A[[X_{α+1}]]) be defined on each vertex v as the ϕ-sum along any path from any vertex of (G_{α+1}−G_{α+1,n_{0}})v_{1,0} to v. Then ψ_{α+1} ∈ C^{0}(A_{G_{α+1}}[[X_{α+1}]]) and ϕ|_{X_{α+1}} = ∂^{∗}(ψ_{α+1}).

The α < ω_{1} which are respected by ϕ converge to ω_{1}, and it follows that the corresponding (unique) ψ_{α+1} converge to an element ψ ∈ C^{0}(A_{G}[[X]]) such that ϕ = ∂^{∗}(ψ), so ϕ ∈ B^{1}(A_{G}[[X]]).

Hence H^{1}(A_{G}[[X]]) = 0, so H^{1}(G, AG) = 0. □

Up until now, in this section, we have given a straightforward topological translation of [6], which we felt illuminated the arguments. We now come to a new result.

5.4 Theorem. If G is locally finite, and |G| = ℵ_{1}, then H^{2}(G, AG) has a subgroup isomorphic to A^{ℵ_{1}}, and | H^{2}(G, AG)| = |A^{G}| = |A|^{ℵ_{1}}.

Proof. Let X be as in Definitions 5.1.

Let α ∈ ω′_{1}.

Let M′_{α} denote the G_{α}-space with G_{α}-transversal [−1, 0]×ℝ^{+}, where the G_{α}-sta-bilizers of v_{−1,n}, y_{−1,n}, x_{−1,n} and f_{−1,n} are G_{α,n}^{•}(= G_{0} ∩ G_{α,n}), while the G_{α}-sta-bilizers of v_{0,n} and y_{0,n} are G_{α,n}. This completely specifies M′_{α}.

Notice that M′_{α} is the mapping cylinder of an injective G_{α}-map G_{α}×_{G_{0}} Y_{α}^{•} → Y_{α}, where Y_{α}^{•} is the G_{0}-tree of Definition 4.4 corresponding to the chain (G_{α,n}^{•} | n < ω_{0}), a G_{0}-subtree of Y_{α}.

We have a cellular G_{α}-map Y_{α} → X_{α}, and, similarly, by Corollary 2.3, we can construct a cellular G_{0}-map Y_{α}^{•} → Y_{0} between spaces of type \underline{E}G_{0}. These two maps can be extended to a G_{α}-map M′_{α} → X_{α}.

Notice that Y_{α} is contained in both M_{α} and M′_{α}, and the map Y_{α} → X_{α} has been extended to M_{α} → X_{α+1} and to M′_{α} → X_{α}.

For each n < ω_{0}, let λ′_{α,n} denote the least ordinal such that the X_{λ′_{α,n}+1} contains the image of the face f_{−1,n}, under the map M′_{α} → X_{α}. Notice that λ′_{α,n} + 1 < α, since α is a limit ordinal. Choose a strictly ascending sequence (λ_{α,n} < α | n < ω_{0}) with limit α, such that λ_{α,n} > max{λ′_{α,i} | 0 ≤ i ≤ n}. Let

h_{α,n} := g_{λ_{α,n}} ∈ G_{λ_{α,n}+1} − G_{λ_{α,n}}.

Let Q = {h_{α,n}f_{α,n} | (α, n) ∈ ω′_{1} × ℕ}, so

A[[Q]] ⊆ A_{G}[[F X]] = C^{2}(A_{G}[[X]]) = Z^{2}(A_{G}[[X]]).

Recall that a subset S of ω_{1} is said to be ω_{1}-stationary if S meets each ω_{1}-club. Let P be a partition of ω′_{1} into ℵ_{1} subsets, each being ω_{1}-stationary; see, for example, [7, Lemma 7.6, p.59].

Consider any ϕ ∈ A^{P}, [α] ↦ ϕ[α].

Define

ϕ^{†}: = ∑_{(α,n)∈ω′_{1}×ℕ} ϕ[α].h_{α,n}f_{α,n} ∈ A[[Q]] ⊆ Z^{2}(A_{G}[[X]]).

It is easy to see that ϕ^{†} respects all the ordinals in ω_{1}.

We claim that if ϕ^{†} ∈ B^{2}(A_{G}[[X]]), then ϕ = 0.

Suppose not, so there exists ψ ∈ C^{1}(A_{G}[[X]]) = A_{G}[[EX]], such that ∂^{∗}ψ = ϕ^{†}, and there exists p ∈ P such that ϕ(p) ≠ 0. We shall obtain a contradiction.

Since Y_{0} is countable, there exists μ < ω_{1} such that supp(ψ) ∩ GY_{0} ⊆ G_{μ}Y_{0}. Since p is ω_{1}-stationary, and the set of ordinals respected by ψ is an ω_{1}-club,

there exists α ∈ p such that α > μ and ψ respects α. Notice ϕ[α] = ϕ(p) ≠ 0.

Now ϕ^{†} and ψ induce elements ϕ_{α+1} and ψ_{α+1}, respectively, in A_{G_{α+1}}[[M_{α}]],

and, moreover, ∂^{∗}ψ_{α+1} = ϕ_{α+1} in A_{G_{α+1}}[[M_{α}]]. Notice ϕ_{α+1} = ∑_{n<ω_{0}} ϕ(p).h_{α,n}f_{0,n}.

Let ξ := − ∑_{n<ω_{0}} ϕ(p).h_{α,n}y_{0,n} ∈ A_{G_{α}}[[EY_{α}]] ⊆ A_{G_{α+1}}[[M_{α}]]. Then ∂^{∗}(ξ) = ϕ_{α+1}

in A_{G_{α+1}}[[M_{α}]]. Thus ∂^{∗}(ψ_{α+1} − ξ) = 0. Moreover, ψ_{α+1} − ξ respects α, since both ξ and ψ_{α+1} respect α. By the proof of Theorem 5.3, there exists ν < α such that the (ψ_{α+1} −ξ)-sum along any path in Y_{α} between any two vertices in (G_{α} −G_{ν})v_{0,0} is zero.

Now ϕ^{†} and ψ induce elements ϕ_{α} and ψ_{α} in A_{G_{α}}[[M′_{α}]]; moreover, ∂^{∗}ψ_{α} = ϕ_{α} in A_{G_{α}}[[M′_{α}]], and ψ_{α+1} and ψ_{α} agree on Y_{α}. Also, ξ can be viewed as an element of A_{G_{α}}[[M′_{α}]], so the (ψ_{α} − ξ)-sum along any path in Y_{α} between any two vertices in (G_{α} − G_{ν})v_{0,0} is zero.

There exists κ < α such that ψ_{α} vanishes on (G_{α} − G_{κ})x_{−1,0}.

Choose n < ω_{0} such that λ_{α,n} is greater than μ, ν and κ. Choose g ∈ G_{α,n+1}^{•} − G_{α,n}^{•}. Thus g ∈ G_{0}, g fixes x_{−1,n+1}, and g moves y_{0,n}.

In ℤ[M_{α}],

∂(∑_{i=0}^{n} f_{−1,i}) = x_{−1,0} − x_{−1,n+1} + ∑_{i=0}^{n}(y_{0,i} − y_{−1,i}).

Hence

∂(h_{α,n}(1 − g) ∑_{i=0}^{n} f_{−1,i}) = h_{α,n}(1 − g)(x_{−1,0} + ∑_{i=0}^{n}(y_{0,i} − y_{−1,i})).

We can view ψ_{α} as an additive map ℤ[EM_{α}] → A, and apply it to the foregoing equation, to get

ϕ_{α}(h_{α,n}(1 − g) ∑_{i=0}^{n} f_{−1,i})

=ψ_{α}(h_{α,n}(1 − g)(x_{−1,0} + ∑_{i=0}^{n}(y_{0,i} − y_{−1,i}))).

Notice that h_{α,n}G_{λ_{α,n}} ∩ G_{λ_{α,n}} = ∅.

Since h_{α,n}, h_{α,n}g ∉ G_{λ′_{α,i}}, for 0 ≤ i ≤ n, we see ϕ_{α}(h_{α,n}(1 − g) ∑_{i=0}^{n} f_{−1,i}) = 0.

Also, h_{α,n}, h_{α,n}g ∉ G_{μ}, so ψ_{α}(h_{α,n}(1 − g)y_{−1,i}) = 0.

Also, h_{α,n}, h_{α,n}g ∉ G_{κ}, so ψ_{α}(h_{α,n}(1 − g)x_{−1,0}) = 0.

It follows that ψ_{α}(h_{α,n}(1 − g) ∑_{i=0}^{n} y_{0,i}) = 0.

But h_{α,n}, h_{α,n}g ∉ G_{ν}, so (ψ_{α} − ξ)(h_{α,n}(1 − g) ∑_{i=0}^{n} y_{0,i}) = 0. Hence

ξ(∑_{i=0}^{n} h_{α,n}(1 − g)y_{0,i}) = 0.

Now ξ(h_{α,n}y_{0,n}) = −ϕ(p), and ξ vanishes on all other summands because h_{α,n}G_{0} ∩ G_{λ_{α,i}+1} = ∅ for 0 ≤ i ≤ n − 1, and gy_{0,n} ≠ y_{0,n}. Hence ϕ(p) = 0, which is a contradiction.

Since |P| = ℵ_{1}, we have an embedding of A^{ℵ_{1}} in H^{2}(G, AG). □

6. Cohomology of directed unions

In this section, we recall some known results about cohomology for well-ordered directed unions, with special emphasis on abelian groups.

6.1 Notation. We let (P(G), ∂) denote the bar resolution for G, and let P_{n}(G) denote its nth component, for each n ∈ ℤ. Thus (P(G), ∂) is a free ℤG-resolution of ℤ, and, for n ≥ 0, P_{n}(G) has as ℤ-basis the Cartesian power G^{n+1}, with G acting by left multiplication on the first coordinate, and, for n ≥ 1,

∂_{n}(g_{0}, …, g_{n}) :=

∑_{i=0}^{n−1}(−1)^{i}(g_{0}, …, g_{i−1}, g_{i}g_{i+1}, g_{i+2}, …, g_{n}) + (−1)^{n}(g_{0}, …, g_{n−1}).

As usual, if n ≤ −1, P_{n}(G) = 0, and, if n ≤ 0, ∂_{n} = 0. □

The following is a degenerate case of the cohomology spectral sequence for well-ordered directed unions; see, for example [13, Section 3].

6.2 Lemma (Robinson [13, Proposition 1]). Let n ∈ ℕ, let M be a ℤG-module, let β be a limit ordinal, and let (G_{α} | α ≤ β) be a continuous chain of subgroups of G. If H^{n−1}(G_{α}, M) = 0 for all α < β, then H^{n}(G_{β}, M) = lim_{\overleftarrow{α<β}} H^{n}(G_{α}, M).

Proof. For each α ≤ β, view the bar resolution P(G_{α}) as a ℤ-subcomplex of P(G_{β}), so (P_{n}(G_{α}) | α ≤ β) is a continuous chain.

We want to show that the natural map

(6.3)  H^{n}(G_{β}, M) → lim_{\overleftarrow{α<β}} H^{n}(G_{α}, M)

is bijective.

We begin by showing it is injective.

Consider any element ξ of the kernel of (6.3). Then ξ is represented by an n-cocycle ϕ_{β}: P_{n}(G_{β}) → M, so ϕ_{β} is ℤG_{β}-linear, and ϕ_{β} ∘ ∂_{n+1} = 0.

Consider any α ≤ β. Let ϕ_{β,α} denote the restriction of ϕ_{β} to P_{n}(G_{α}). We shall construct, transfinitely, a continuous directed system of maps

(ψ_{β,α}: P_{n−1}(G_{α}) → M | α ≤ β)

such that ψ_{β,α} is ℤG_{α}-linear, and ψ_{β,α} ∘ ∂_{n} = ϕ_{β,α}. It will then follow that ϕ_{β} = ϕ_{β,β} = ψ_{β,β} ∘ ∂_{n} is a coboundary, that ξ = 0, and that (6.3) is injective.

If α < β, then ϕ_{β,α} represents an element ξ_{α} of H^{n}(G_{α}, M), and, since ξ lies in the kernel of (6.3), ξ_{α} = 0, so there exists a ℤG_{α}-linear map

ψ_{α}: P_{n−1}(G_{α}) → M

such that ψ_{α} ∘ ∂_{n} = ϕ_{β,α}.

Let ψ_{β,0} = ψ_{0}.

If α < β and we have constructed ψ_{β,α}, then we construct ψ_{β,α+1} as follows. Let ψ_{α+1,α} denote the restriction of ψ_{α+1} to P_{n−1}(G_{α}). Since ψ_{α+1} ∘ ∂_{n} = ϕ_{β,α+1}, we see that ψ_{α+1,α} ∘∂_{n} = ϕ_{β,α} = ψ_{β,α} ∘∂_{n}. Thus ψ_{β,α} −ψ_{α+1,α} is an (n−1)-cocycle, so represents an element of H^{n−1}(G_{α}, M). By hypothesis, this element is zero, so there exists a ℤG_{α}-linear map μ_{α}: P_{n−2}(G_{α}) → M such that ψ_{β,α} − ψ_{α+1,α} = μ_{α} ∘ ∂_{n−1}. Let μ_{α,α+1}: P_{n−2}(G_{α+1}) → M denote the unique ℤG_{α+1}-linear map which is μ_{α} on P_{n−2}(G_{α}), and is zero on G_{α+1}^{n−1} − (G_{α+1} × G_{α}^{n−2}). Now define

ψ_{β,α+1}: = ψ_{α+1} + μ_{α,α+1} ∘ ∂_{n−1}.

By construction, ψ_{β,α+1} acts on P_{n−1}(G_{α}) as ψ_{α+1,α} + μ_{α} ∘ ∂_{n−1} = ψ_{β,α}. Also ψ_{β,α+1} ∘ ∂_{n} = ψ_{α+1}∂_{n} + 0 = ϕ_{β,α+1}.

If α is a limit ordinal in β + 1 and we have constructed (ψ_{β,α′} | α′ < α), then the latter has a direct limit which we take to be ψ_{β,α}.

This completes the construction, so (6.3) is injective.

A similar, but easier, argument shows that (6.3) is surjective, and here the hypothesis on H^{n−1} is not needed. □

A special case of this result appeared in the penultimate paragraph of the proof of Theorem 5.3. We can now say even more.

6.4 Theorem (Holt [6]). If G is locally finite, and ℵ-rank(G) ≠ 0, then H^{1}(G, AG) = 0.

Proof. By (1.3), we may assume that ℵ-rank(G) ≥ 1, and we proceed by induction on ℵ-rank(G). If ℵ-rank(G) = 1, the assertion holds by Theorem 5.3. Thus we may assume that ℵ-rank(G) ≥ 2 and the result holds for smaller groups. There exists an ordinal β and a continuous chain (G_{α} | α ≤ β) of subgroups of G with G_{β} = G, and 1 ≤ ℵ-rank(G_{α}) < ℵ-rank(G) for all α < β.

By (1.3) and the induction hypothesis, H^{0}(G_{α}, AG) = H^{1}(G_{α}, AG) = 0 for all α < β, since AG is an induced ℤG_{α}-module. By Lemma 6.2, H^{1}(G, AG) = 0. □

6.5 Remark. In light of Theorem 3.10, the method of proof of Theorem 6.4 shows that

“If n ∈ ℕ, G is locally finite, and n ≠ ℵ-rank(G) + 1, then H^{n}(G, ℤG ⊗_{ℤ} −) = 0, ”

is equivalent to

“If n ∈ ℕ, G is locally finite, and n = ℵ-rank(G), then H^{n}(G, ℤG ⊗_{ℤ} −) = 0.” □

6.6 Definitions. Let H be a subgroup of G, and let L, M be ℤG-modules.

Let N_{G}(H) denote the normalizer of H in G, and let C_{G}(H) denote the centralizer of H in G.

There is a natural action of N_{G}(H) on Ext_{ℤH}^{∗}(L, M), where L, M are viewed as ℤH-modules by restriction. Perhaps the simplest way to define this action is as follows. Let (P, ∂) be a projective ℤG-resolution of L, and view (P, ∂) as a projective ℤH-resolution of L. Then any g ∈ N_{G}(H) gives rise to an action on Hom_{ℤH}(P, M), ϕ ↦ ϕ^{g}, where ϕ^{g}(p) = g^{−1}(ϕ(gp)). Since g normalizes H, we see that ϕ^{g} is ℤH-linear. The N_{G}(H)-action respects cocycles and coboundaries, so induces an action on the cohomology, Ext_{ℤH}^{∗}(L, M).

In the foregoing, if ϕ is ℤG-linear, then ϕ^{g} = ϕ. It follows that the image of the restriction map

Ext_{ℤG}^{∗}(L, M) → Ext_{ℤH}^{∗}(L, M)

is contained in the set Ext_{ℤH}^{∗}(L, M)^{N_{G}(H)} of points fixed by N_{G}(H).

Let g be an element of C_{G}(H). Left multiplication by g determines a ℤH-linear endomorphism of M and we denote it by g·|_{M}. Similarly, g·|_{P} ∈ End_{ℤH}(P) is a lift of g·|_{L} ∈ End_{ℤH}(L). Now Ext_{ℤH}^{∗}(L, M) is a bimodule over End_{ℤH}(M) and End_{ℤH}(L), and we see that the action of g on Ext_{ℤH}^{∗}(L, M) is given by η ↦ (g·|_{M})^{−1}∘η∘(g·|_{L}).

If we restrict to the case where L = ℤ with trivial G-action, we see that the image of the restriction map H^{∗}(G, M) → H^{∗}(H, M) is contained in H^{∗}(H, M)^{N_{G}(H)}, and hence in H^{∗}(H, M)^{C_{G}(H)}, and here C_{G}(H) acts by left multiplication on M. □

The following is now an immediate consequence of Lemma 6.2

6.7 Corollary. Let n ∈ ℕ, let M be a ℤG-module, let β be a limit ordinal, and let (G_{α} | α ≤ β) be a continuous chain of subgroups of G with G_{β} = G. If, for each α < β, H^{n}(G_{α}, M)^{N_{G}(G_{α})} = 0 and H^{n−1}(G_{α}, M) = 0, then H^{n}(G, M) = 0. □

We record consequences for abelian groups which seem to be new.

6.8 Theorem. Let G be an abelian group, λ an ordinal, and Δ a G-set with stabilizers of ℵ-rank strictly less than λ.

(1) For each n ∈ ℕ, if ℵ-rank(G) ≥ λ + n then H^{n}(G, AΔ) = 0.

(2) If ℵ-rank(G) ≥ λ + ω_{0} then H^{∗}(G, AΔ) = 0.

Proof. (1). We argue by induction on n.

If n = 0, then all G-stabilizers of elements of Δ have infinite index in G, so Δ has no finite G-orbits. Thus (AΔ)^{G} = 0, that is, H^{0}(G, AΔ) = 0.

Now suppose that n ≥ 1, and that the result holds for smaller n. Let β denote the least ordinal of cardinality rank(G), so β is a limit ordinal. Moreover, there exists a continuous chain of subgroups (G_{α} | α ≤ β) such that G_{β} = G, and, for each α < β,

λ + n − 1 ≤ ℵ-rank(G_{α}) < ℵ-rank(G).

Consider α < β.

By the induction hypothesis, H^{n−1}(G_{α}, AΔ) = 0, so, by Corollary 6.7, it remains to show that H^{n}(G_{α}, AΔ)^{N_{G}(G_{α})} = 0. Here N_{G}(G_{α}) = C_{G}(G_{α}) = G, since G is abelian. Thus, we want to show that H^{n}(G_{α}, AΔ)^{G} = 0 where G is acting via multiplication on AΔ.

Consider any element ζ ∈ H^{n}(G_{α}, AΔ). We can use the bar resolution P(G_{α}), and represent ζ by a ℤG_{α}-linear map ϕ: P_{n}(G_{α}) → AΔ. Since |G_{α}| < |G|, we see there is a G_{α}-subset Δ′ of Δ such that AΔ′ contains the image of ϕ, and |Δ′| < |G|. Let Δ′′ be the complement of Δ′ in Δ. Then AΔ = AΔ′ ⊕ AΔ′′, so

H^{n}(G_{α}, AΔ) = H^{n}(G_{α}, AΔ′) ⊕ H^{n}(G_{α}, AΔ′′),

and, in the corresponding expression ζ = (ζ′, ζ′′), we have ζ′′ = 0.

Now

|{g ∈ G | gΔ′ ∩ Δ′ ≠ ∅}| < |G|

because the elements of Δ have G-stabilizers of cardinality strictly less than |G|. Hence there exists g ∈ G such that gΔ′ ∩ Δ′ = ∅, that is, gΔ′ ⊆ Δ′′, so

H^{n}(G_{α}, AΔ′)^{g} ⊆ H^{n}(G_{α}, AΔ′′).

It follows that if ζ^{g} = ζ then ζ = 0. This proves that H^{n}(G_{α}, AΔ)^{G} = 0, and (1) is proved.

(2) follows from (1). □

The case where Δ = G is of interest; here we can take λ = 0.

6.9 Corollary. Let n ∈ ℕ, and let G be an abelian group. Then, H^{n}(G, AG) = 0 if n < ℵ-rank(G) + 1. □

6.10 Corollary. Let n ∈ ℕ, and let G be an abelian, torsion group.

(1) (Holt [5]). H^{n}(G, AG) = 0 if n ≠ ℵ-rank(G) + 1.

(2) (Chen [12, Corollary 7.6]). If R is a nonzero, o(G)-inverting ring, then cd_{R} G = n if and only if ℵ-rank(G) = n − 1, that is,

cd_{R} G = min{ℵ-rank(G) + 1, ∞}. □

Proof. (1) follows from Theorem 3.10 and Corollary 6.9. (2) follows from the fact that, if cd_{R}(G) < ∞, then H^{m}(G, ℤG ⊗_{ℤ} −) ≠ 0 for m = cd_{R}(G); see the Commentary on Conjecture 1.7. □

7. Cardinals, free abelian groups, and H 𝔉

We now recall the hierarchies introduced in [9]; see [10] for more details.

7.1 Notation. Let 𝔛 denote a class of groups.

All the classes of groups that we consider are closed under isomorphism, for example, the class 𝔉 of all finite groups.

We let L 𝔛 denote the class of groups whose finitely generated subgroups all lie in 𝔛. For example, if 𝔛 contains all finitely generated abelian groups, then L 𝔛 contains all abelian groups.

We let H_{1} 𝔛 denote the class of all groups G for which there exists a finite-di-mensional contractible G-complex with all stabilizers lying in 𝔛. For example, H_{1} 𝔉 contains all finitely generated abelian groups, since, if G is finitely-generated and abelian, then G has a finite subgroup N such that G/N is isomorphic to ℤ^{n} for some n ∈ ℕ, and thus G/N acts freely on ℝ^{n} preserving a CW-structure.

If H_{1} 𝔛 = 𝔛, then 𝔛 is said to be H_{1}-closed.

We let H 𝔛 denote the smallest H_{1}-closed class of groups which contains 𝔛. This class has a hierarchy indexed by the ordinals, where for each ordinal β, we define H_{β} 𝔛 recursively, by setting

H_{0} 𝔛 : = 𝔛,
H_{β} 𝔛 : = H_{1} H_{β−1} 𝔛 if β is a successor ordinal,
H_{β} 𝔛 : = ⋃_{α<β} H_{α} 𝔛 if β is a limit ordinal. □

We can use Theorem 2.8 to get new sufficient conditions for membership in H 𝔛.

7.2 Theorem. Let 𝔛 be a subgroup-closed class of groups, and let G ∈ L 𝔛. (1) If ℵ-rank(G) = −1 then G ∈ H_{0} 𝔛.

(2) If ℵ-rank(G) < ω_{0} then G ∈ H_{1} 𝔛.

(3) If ℵ-rank(G) = ω_{0} then G ∈ H_{2} 𝔛.

Proof. (1). If ℵ-rank(G) = −1 then G is a finitely generated element of L 𝔛, so lies in 𝔛 = H_{0} 𝔛.

(2). If ℵ-rank(G) = n for some n ∈ ℕ then, by Theorem 2.8, G acts on a contractible (n + 1)-dimensional CW-complex with stabilizers contained in finitely generated subgroups of G. Since G lies in L 𝔛, and 𝔛 is subgroup closed, we see that these stabilizers lie in 𝔛. Hence G ∈ H_{1} 𝔛.

(3). If ℵ-rank(G) = ω_{0}, then we can write G as the union of an ascending chain (G_{n} | n ∈ ℕ) of subgroups, such that, for each n ∈ ℕ, ℵ-rank(G_{n}) = n. As in Remark 2.5, there exists a G-tree with each cell stabilizer contained in G_{n}, for some n ∈ ℕ, so lying in H_{1} 𝔛 by (2). Hence, G ∈ H_{2} 𝔛. □

We next consider some necessary conditions for membership in H 𝔛, which are natural generalizations of [8, Lemma 1].

7.3 Lemma. Let n ∈ ℕ, let R be a ring, let

0 → M_{n} → M_{n−1} → ⋯ → M_{1} → M_{0} → M_{−1} → 0

be an exact sequence of R-modules, and let L be an R-module.

Suppose that Ext_{R}^{i}(L, M_{i}) = 0 for i = 0, …, n. Then Ext_{R}^{0}(L, M_{−1}) = 0.

Proof. Clearly the result holds for n = 0. Thus we may assume that n ≥ 1, and that the result holds with n − 1 in place of n.

Let M′_{n−1} denote the cokernel of the map M_{n} → M_{n−1}, so we have exact sequences

(7.4)  0 → M_{n} → M_{n−1} →M′_{n−1} → 0
(7.5)  0 →M′_{n−1} → M_{n−2} → ⋯ → M_{0} → M_{−1} → 0

Now (7.4) gives rise to a long exact sequence which contains the segment

Ext_{R}^{n−1}(L, M_{n−1}) → Ext_{R}^{n−1}(L, M′_{n−1}) → Ext_{R}^{n}(L, M_{n}).

Here the outer terms are zero, by hypothesis, so the inner term is zero. The induction hypothesis can now be applied to (7.5), and we see that Ext_{R}^{0}(L, M_{−1}) = 0.

The result follows. □

We record the contrapositive of the case where R = ℤG, and M_{−1} = L = ℤ with trivial G-action.

7.6 Corollary. If n ∈ ℕ, and

0 → M_{n} → M_{n−1} → ⋯ → M_{1} → M_{0} → ℤ → 0

is an exact sequence of ℤG-modules, then there exists i such that 0 ≤ i ≤ n and H^{i}(G, M_{i}) ≠ 0. □

7.7 Proposition. If 𝔛 is a class of groups, and H^{∗}(G, ℤΔ) = 0 for every G-set Δ for which all stabilizers lie in 𝔛, then G ∉ H_{1} 𝔛.

Proof. Suppose that G ∈ H_{1} 𝔛, so there exists a finite-dimensional, contractible CW-complex X on which G acts with all stabilizers lying in 𝔛. Let n denote the dimension of X. The augmented cellular chain complex of X,

0 → C_{n}(X) → C_{n−1}(X) → ⋯ → C_{1}(X) → C_{0}(X) → ℤ → 0,

is an exact sequence of ℤG-modules, so, by Corollary 7.6, there exists i such that 0 ≤ i ≤ n and H^{i}(G, C_{i}(X)) ≠ 0. Thus H^{∗}(G, C_{i}(X)) ≠ 0. But we can write C_{i}(X) = ℤΔ, where Δ is the set of i-dimensional open cells of X, so is a G-set with all stabilizers in 𝔛. This contradicts the hypothesis. □

Combining Theorem 6.8(2) and Proposition 7.7, we get the following.

7.8 Corollary. Let 𝔛 be a class of groups and λ be an ordinal. Suppose that G is an abelian group in H_{1} 𝔛 such that every subgroup of G which lies in 𝔛 has ℵ-rank strictly less than λ. Then ℵ-rank(G) < λ + ω_{0}. □

7.9 Theorem. If β is any ordinal, then every abelian group in H_{β} 𝔉 has ℵ-rank strictly less than ω_{0}β.

Proof. We argue by induction on β.

The result holds for β = 0 by definition of 𝔉.

Thus we may assume that β > 0, and that the result holds for smaller ordinals. Consider the case where β is a limit ordinal. Here, each abelian group in H_{β} 𝔉

lies in H_{α} 𝔉 for some α < β, so, by the induction hypothesis, is of cardinality strictly less than ℵ_{ω_{0}α} ≤ ℵ_{ω_{0}β}.

Now consider the case where β is a successor ordinal, and write β = α + 1. By the induction hypothesis, every abelian group in H_{α} 𝔉 is of cardinality strictly less than ℵ_{ω_{0}α}. If we apply Corollary 7.8, with 𝔛 = H_{α} 𝔉, we see that every abelian group in H_{1} 𝔛 = H_{β} 𝔉 is of cardinality strictly less than ℵ_{ω_{0}α+ω_{0}} = ℵ_{ω_{0}β}. □

We can now make six statements, of varying profundity and novelty, about how free abelian groups fit into the hierarchy H 𝔉.

7.10 Theorem. For each cardinal κ, let 𝔸_{κ} denote the free abelian group of rank κ. (1) 𝔸_{0} ∈ H_{0} 𝔉.

(2) (𝔸_{κ} | 1 ≤ κ < ℵ_{0}) ⊆ H_{1} 𝔉 − H_{0} 𝔉.

(3) (𝔸_{κ} | ℵ_{0} ≤ κ < ℵ_{ω_{0}}) ⊆ H_{2} 𝔉 − H_{1} 𝔉.

(4) 𝔸_{ℵ_{ω_{0}}} ∈ H_{3} 𝔉 − H_{2} 𝔉.

(5) For each finite ordinal n, 𝔸_{ℵ_{ω_{0}n}} ∉ H_{n+1} 𝔉; equivalently, every free abelian group in H_{n+1} 𝔉 has ℵ-rank strictly less than ω_{0}n.

(6) For each infinite ordinal β, 𝔸_{ℵ_{ω_{0}β}} ∉ H_{β} 𝔉.

Proof. (6) is a special case of Theorem 7.9.

(1) is clear.

(2). Consider n ∈ ℕ. Then 𝔸_{n} acts freely on an n-dimensional CW-complex with underlying space ℝ^{n}, so 𝔸_{n} ∈ H_{1} 𝔉.

It follows that 𝔸_{n} ∈ H_{1} 𝔉, and it is clear that, if n ≥ 1, then 𝔸_{n} ∉ H_{0} 𝔉.

(5). It is well known that, for each n ∈ ℕ, H^{n}(𝔸_{n}, ℤ) = ℤ. This can be seen by induction, using a long exact sequence in cohomology which gives a short exact sequence of graded groups

0 → H^{∗}(𝔸_{n−1}, ℤ) → H^{∗+1}(𝔸_{n}, ℤ) → H^{∗+1}(𝔸_{n−1}, ℤ) → 0;

it can also be seen from the fact that the n-torus is a K(𝔸_{n}, 1).

Suppose that 𝔸_{ℵ_{0}} ∈ H_{1} 𝔉, so, for some n ∈ ℕ, 𝔸_{ℵ_{0}} acts freely on a contractible n-dimensional CW-complex X. Hence 𝔸_{n+1} acts freely on X, so H^{n+1}(𝔸_{n+1}, −) = 0, which contradicts the fact that H^{n+1}(𝔸_{n+1}, ℤ) = ℤ. Thus 𝔸_{ℵ_{0}} ∉ H_{1} 𝔉, so every free abelian group in H_{1} 𝔉 has finite rank, that is, has ℵ-rank equal to −1.

Now suppose that n ≥ 1, and that every free abelian group in H_{n} 𝔉 has ℵ-rank strictly less than ω_{0}(n − 1). Corollary 7.8, with 𝔛 = H_{n} 𝔉, and λ = ω_{0}(n − 1), implies that every free abelian group in H_{n+1} 𝔉 has ℵ-rank strictly less than ω_{0}n.

Now (5) has been proved by induction.

(3) and (4). By (2), H_{1} 𝔉 includes all finitely generated free abelian groups. Hence every free abelian group lies in L H_{1} 𝔉. If we apply Theorem 7.2(2) and (3), with 𝔛 = H_{1} 𝔉, we find that the free abelian groups of ℵ-rank strictly less than ω_{0} lie in H_{2} 𝔉, and that the free abelian group of ℵ-rank ω_{0} lies in H_{3} 𝔉. Combined with (5), these imply (3) and (4). □

7.11 Remark. Theorem 7.10 gives an interesting new proof that

H_{0} 𝔉 ≠ H_{1} 𝔉 ≠ H_{2} 𝔉 ≠ H_{3} 𝔉. □

7.12 Conjecture. 𝔸_{ℵ_{ω_{0}+1}} ∉ H_{3} 𝔉; equivalently, 𝔸_{ℵ_{ω_{0}+1}} ∉ H 𝔉. □

7.13 Conjecture. H_{3} 𝔉 ≠ H 𝔉. □

7.14 Conjecture. There exists an ordinal α such that H_{α} 𝔉 = H 𝔉. □

Acknowledgments. W. Dicks was partially supported by the DGES and the DGI through grants PB96-1152 and BFM2000-0354, respectively. I. Leary was partially supported by the EPSRC. S. Thomas was partially supported by NSF grants.

I. Leary and P. Kropholler thank the Centre de Recerca Matemàtica of the Institut d’Estudis Catalans for the hospitality they received, and S. Thomas thanks the Departament de Matemàtiques of the Universitat Autònoma de Barcelona for its hospitality.

References

1. M. Auslander, On the dimension of modules and algebras, III, Nagoya Math. J. 9 (1955), 67–77.

2. Kenneth S. Brown, Cohomology of groups, GTM 87, Springer-Verlag, Berlin, 1982.

3. László Fuchs, Infinite abelian groups, Vol. I, Academic Press, New York, 1970.

4. D. F. Holt, On the cohomology of locally finite groups, Quart. J. Math. Oxford 32 (1981), 165–172.

5. D. F. Holt, The cohomological dimensions of locally finite groups, J. London Math. Soc. 24 (1981), 129-134.

6. D. F. Holt, Uncountable locally finite groups have one end, Bull. London Math. Soc. 13 (1981), 557-560.

7. T. Jech, Set theory, Academic Press, New York, 1978.

8. P. H. Kropholler, Soluble groups of type FP_{∞} have finite torsion-free rank, Bull. London Math. Soc. 25 (1993), 558–566.

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

10. P. H. Kropholler and G. Mislin, Groups acting on finite dimensional spaces with finite stabilizers, Comment. Math. Helv. 73 (1998), 122–136.

11. P. H. Kropholler and S. Thomas, The consistency of Holt’s conjectures on cohomological dimension of locally finite groups, J. London Math. Soc. 55 (1997), 76–86.

12. B. Osofsky, Projective dimensions of “nice” directed unions, J. Pure Appl. Algebra 13 (1978), 179–219.

13. D. J. S. Robinson, Vanishing theorems for cohomology of locally nilpotent groups, pp. 120–129 in: Group Theory, Proceedings, Brixen/Bressanone 1986, (Eds. O.H. Kegel, F. Mengazzo, G. Zacher), LNM 1281, Springer–Verlag, Berlin, 1987.

14. Simon Thomas, An independence result in group cohomology, Bull. London Math. Soc. 28 (1996), 264–268.

15. Tammo tom Dieck, Transformation groups, Studies in Mathematics 8, de Gruyter, Berlin, 1987.

Departament de Matemàtiques, Universitat Autònoma de Barcelona, E-08193 Bellaterra (Barcelona), Spain

E-mail address: dicks@mat.uab.es

School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS, United Kingdom

E-mail address: P.H.Kropholler@qmw.ac.uk

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

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

Department of Mathematics, Rutgers University, New Brunswick, N. J. 08903, USA

E-mail address: sthomas@math.rutgers.edu