ON GROUPS ACTING ON CONTRACTIBLE SPACES WITH STABILIZERS OF PRIME POWER ORDER

IAN J. LEARY AND BRITA E. A. NUCINKIS

Abstract. Let 𝔉 denote the class of finite groups, and let 𝔓 denote the subclass consisting of groups of prime power order. We study group actions on topological spaces in which either (1) all stabilizers lie in 𝔓 or (2) all stabilizers lie in 𝔉. We compare the classifying spaces for actions with stabilizers in 𝔉 and 𝔓, the Kropholler hierarchies built on 𝔉 and 𝔓, and group cohomology relative to 𝔉 and to 𝔓. In terms of standard notations, we show that 𝔉 ⊂ H_{1}𝔓 ⊂ H_{1}𝔉, with all inclusions proper; that H𝔉 = H𝔓; that 𝔉H^{∗}(G; −) = 𝔓H^{∗}(G; −); and that E_{𝔓}G is finite-dimensional if and only if E_{𝔉}G is finite-dimensional and every finite subgroup of G is in 𝔓.

1. Introduction

Let F denote a class of groups, by which we mean a collection of groups which is closed under isomorphism and taking subgroups. A G-CW-complex X is said to be a model for E_{F}G, the classifying space for actions of G with stabilizers in F, if the fixed point set X^{H} is contractible for H ∈ F and is empty for H ∉ F. The most common classes considered are the class of trivial groups and the class 𝔉 consisting of all finite groups. In these cases E_{F}G is often denoted EG and \underline{E}G respectively. Note that EG is the total space of the universal principal G-bundle, or equivalently the universal covering space of an Eilenberg-Mac Lane space for G. The space \underline{E}G is called the classifying space for proper actions of G. Recently there has been much interest in finiteness conditions for the spaces E_{F}G, especially for \underline{E}G. Milnor and Segal’s constructions of EG both generalize easily to construct models for any E_{F}G, and one can show that any two models for E_{F}G are naturally equivariantly homotopy equivalent.

For some purposes the structure of the fixed point sets for subgroups in F is irrelevant. For example, a group is in Kropholler’s class H_{1}F if there is any finite-dimensional contractible G-CW-complex X with all stabilizers in F. The class H_{1}F is the first stage of a hierarchy whose union is Kropholler’s class HF of hierarchically decomposable groups [10]. (These definitions were first considered for the class 𝔉 of all finite groups, but work for any class F.)

The first named author was partially supported by NSF grant DMS-0804226 and by the Heilbronn Institute.

A priori, the class H_{1}F may contain groups G that do not admit a finite-dimensional model for E_{F}G, and we shall give such examples in the case when F = 𝔓, the class of groups of prime power order. By contrast, in the case when F = 𝔉, no group G is known to lie in H_{1}𝔉 without also admitting a finite-dimensional model for \underline{E}G. A construction due to Serre shows that every group G in H_{1}𝔉 that is virtually torsion-free has a finite-dimensional \underline{E}G [4], and the authors have given examples of G for which the minimal dimension of a contractible G-CW-complex is lower than the minimal dimension of a model for \underline{E}G [14]. These examples G also have the property that they admit a contractible G-CW-complex with finitely many orbits of cells, but that they do not admit any model for \underline{E}G with finitely many orbits of cells.

Throughout this paper, 𝔉 will denote the class of finite groups, and 𝔓 will denote the class of finite groups of prime power order. We compare the classifying space for G-actions with stabilizers in 𝔓 with the more well-known \underline{E}G, and we compare the Kropholler hierarchies built on 𝔉 and 𝔓. We show that a finite group G that is not of prime power order cannot admit a finite-dimensional E_{𝔓}G, but that every finite group is in H_{1}𝔓. We also construct a group that is in H_{1}𝔉 but not in H_{1}𝔓, and we show that H𝔓 = H𝔉.

In the final section we shall contrast this with cohomology relative to all finite subgroups. The relative cohomological dimension can be viewed as a generalisation of the virtual cohomological dimension, since for virtually torsion free groups these are equal, see [17]. By a result of Bouc [2, 12] it follows that groups belonging to H_{1}𝔉 have finite relative cohomological dimension, but the converse it not known. In contrast to our results concerning classifying spaces, we show that cohomology relative to subgroups in 𝔉 is naturally isomorphic to cohomology relative to subgroups in 𝔓.

2. Classifying spaces for actions with stabilizers in 𝔓

Theorem 2.1. Let G be a finite group. Then G has a finite dimensional model for E_{𝔓}G if and only if G has prime power order.

Proof. If G has prime power order, then a single point may be taken as a model for E_{𝔓}G. Now let G be an arbitrary finite group, let p be a prime dividing the order of G, and assume that there is a p-subgroup P < G, such that N_{G}(P) is not a p-group. Then the Weyl-group W P = N_{G}(P)/P contains a subgroup H of order prime to p. Assume G has a finite dimensional model for E_{𝔓}G, X say. Then the augmented cellular chain complex of the P -fixed point set, X^{P}, is a finite length resolution of ℤ by free H-modules. This gives a contradiction, since ℤ has infinite projective dimension as an H-module for any non-trivial finite group H.

Therefore we may suppose that G is not in 𝔓 and for all non-trivial subgroups P ∈ 𝔓, the normalizer N_{G}(P) is also in 𝔓. Now let N be a minimal normal subgroup of G. This cannot lie in 𝔓 and hence has the same properties as G. Thus, by minimality we may assume N = G and G is simple.

Choose two distinct Sylow p-subgroups P and Q of G, such that their intersection, I say, is of maximal order, and assume that I is non-trivial. Now, the normalizers N_{P} (I) and N_{Q}(I) contain I as a proper subgroup. Also, the group ⟨N_{P} (I), N_{Q}(I)⟩ does not contain P and Q and neither does N_{G}(I) ≥ ⟨N_{P} (I), N_{Q}(I)⟩, which is a p-group by assumption. Hence there exists a Sylow p-subgroup R containing N_{G}(I) and R ∩ P ≥ N_{P} (I). Thus |R ∩ P | > |I| = |P ∩ Q|, which contradicts the maximality of I. Therefore we may assume that in G all Sylow p-subgroups intersect trivially. In such a group we have, for P a Sylow p-subgroup:

H^{∗}(G, 𝔽_{p}) ≅ H^{∗}(P, 𝔽_{p}),

see for example [4, Theorem III.10.3].

Any non-trivial p-group has non-trivial abelianization, and hence H^{1}(P, 𝔽_{p}), which is naturally isomorphic to Hom(P, 𝔽_{p}), is non-trivial. But this implies that H^{1}(G, 𝔽_{p}) ≅ Hom(G, 𝔽_{p}) is non-trivial, and so G admits a surjective homomorphism to a group of order p. Since G is not in 𝔓, it follows that G cannot be simple, which gives the final contradiction. □

Remark 2.2. Yoav Segev pointed out that the above proof could be shortened by quoting the Frobenius normal p-complement theorem [7, 5.26].

Corollary 2.3. For a group G, the following are equivalent. (i) G admits a finite-dimensional E_{𝔓}G; (ii) Every finite subgroup of G is in 𝔓 and G admits a finitedimensional \underline{E}G. □

Remark 2.4. We conclude the section with a remark on the type of E_{𝔓}G. It can proved analogously to Lück’s proof for \underline{E}G [15] that a group G admits a finite type model for E_{𝔓}G if and only if G has finitely many conjugacy classes of groups of prime power order and the Weyl-groups N_{G}(P)/P for all subgroups P of prime power order are finitely presented and of type FP_{∞}. Hence any group admitting a finite type \underline{E}G also admits a finite type E_{𝔓}G. Recall that a finite extension of a group admitting a finite model for EG always has finitely many conjugacy classes of subgroups of prime power order [4, IX.13.2]. Hence the groups exhibited in [14, Example 7.4] are groups admitting a finite type E_{𝔓}G which do not admit a finite type \underline{E}G.

This behaviour is in stark contrast to that of E_{VC}G, the classifying space with virtually cyclic isotropy. Any group admitting a finite dimensional model for E_{VC}G admits a finite dimensional model for \underline{E}G, see [16] and the converse also holds for a large class of groups including all polycyclic-by-finite and all hyperbolic groups [8, 16]. Furthermore, any group admitting a finite type model for E_{VC}G also admits a finite type model for \underline{E}G [9], but it is conjectured [8] that any group admitting a finite model for E_{VC}G has to be virtually cyclic. This has been shown for a class of groups containing all hyperbolic groups [8] and for elementary amenable groups [9].

3. The hierarchies H𝔉 and H𝔓

Proposition 3.1. Let X be a finite dimensional contractible G-CW-complex such that all stabilizers are finite. If there is a bound on the orders of the stabilizers then there exists a finite dimensional contractible G-CW-complex Y and an equivariant map f : Y → X such that Y^{H} = ∅ if H is not a p-group.

Proof. Using the equivariant form of the simplicial approximation theorem, we may assume that X is a simplicial G-CW-complex. To simplify notation the phrase ‘G-space’ shall mean ‘simplicial G-CW-complex’ and ‘G-map’ will mean ‘G-equivariant simplicial map’ throughout the rest of the proof. The space Y will be a G-space in this sense and the map f : Y → X will be a G-map in this sense. The G-space Y is constructed in two stages. Firstly, for each finite H ≤ G we build a finite-dimensional contractible H-space Y_{H} with the property that all simplex stabilizers in Y_{H} lie in 𝔓.

Suppose for now that each such H-space Y_{H} has been constructed. Using the G-equivariant form of the construction used in [11, Section 8] the space Y is constructed as follows. Let I be an indexing set for the G-orbits of vertices in X. For each i ∈ I, let v_{i} be a representative of the corresponding orbit, and let H_{i} be the stabilizer of v_{i}. Let X^{0} denote the 0-skeleton of X. Define a G-space Y^{0} by

Y^{0} = ∐_{i∈I} G ×_{H_{i}} Y_{H_{i}},

and define a G-map f : Y^{0} → X^{0} by f(g, y) = g.v_{i} for all i ∈ I, for all g ∈ G and for all y ∈ Y_{H_{i}}. For each vertex w of X, let Y (w) = f^{−1}(w) ⊂ Y^{0}. Each Y (w) is a contractible subspace of Y^{0}, and the stabilizer of w acts on Y (w).

Now for σ = (w_{0}, …, w_{n}) an n-simplex of X, define a space Y (σ) as the join

Y (σ) = Y (w_{0}) ∗ Y (w_{1}) ∗ ⋯ ∗ Y (w_{n}).

Each vertex of Y (σ) is already a vertex of one of the Y (w_{i}), and so the map f : Y^{0} → X^{0} defines a unique simplicial map f : Y (σ) → σ. By construction, whenever τ is a face of σ, the space Y (τ) is identified with a subspace of Y (σ). This allows us to define Y and f : Y → X as the colimit over the simplices σ of X of the subspaces Y (σ), and to define f : Y → X, which is a G-map of G-spaces. Since each Y (σ) is contractible, it follows that f is a homotopy equivalence, and hence Y is also contractible (see [11, Corollary 8.6]).

It remains to build the H-space Y_{H} for each finite group H < G. In the case when H ∈ 𝔓 we may take a single point to be Y_{H}, and so we may suppose that H ∉ 𝔓. Fix such a subgroup H, and suppose that we are able to construct a finite-dimensional contractible H-space Z_{H} in which each stabilizer is a proper subgroup of H. We may assume by induction that for each K < H we have already constructed the K-space Y_{K}. The H-space Y_{H} can now be constructed from Z_{H} and the spaces Y_{K} using a process similar to the construction of Y from X and the spaces Y_{H}. It remains to construct the H-space Z_{H}.

An explicit construction of an H-space Z_{H} with the required properties is given in [13]. We therefore provide only a sketch of the argument. We may assume that H is not in 𝔓. Let S be the unit sphere in the reduced regular complex representation of H, so that S is a topological space with H-action such that the stabilizer of every point of S is a proper subgroup of H. Since H is not in 𝔓, there are H-orbits in S of coprime lengths. Using this property, it can be shown that the sphere S admits an H-equivariant self-map g : S → S of degree zero. The H-space Z_{H} is defined to be the infinite mapping telescope (suitably triangulated) of the map g. □

Corollary 3.2. If G is in H_{1}𝔉 and there is a bound on the orders of the finite subgroups of G, then G is in H_{1}𝔓. □

Remark 3.3. In Proposition 3.1, the bound on the orders of the stabilizers of X is used only to give a bound on the dimensions of the spaces Y_{H}. In Theorem 3.8 we shall show that H_{1}𝔉 ≠ H_{1}𝔓.

Remark 3.4. The construction in Proposition 3.1 does not preserve cocompactness, because for most finite groups H, the space Y_{H} used in the construction cannot be chosen to be finite. A result similar to Proposition 3.1 but preserving cocompactness can be obtained by replacing 𝔓 by a larger class 𝔒 of groups. Here 𝔒 is defined to be the class of 𝔓-by-cyclic-by-𝔓-groups. A theorem of Oliver [20] implies that any finite group H that is not in 𝔒 admits a finite contractible H-CW-complex Z′_{H} in which all stabilizers are proper subgroups of H. Applying the same argument as in the proof of Proposition 3.1, one can show that given any contractible G-CW-complex X with all stabilizers in 𝔉, there is a contractible G-CW-complex Y′ with all stabilizers in 𝔒 and a proper equivariant map f′ : Y′ → X. (By proper, we mean that the inverse image of any compact subset of X is compact.)

For X a G-CW-complex with stabilizers in 𝔉, and p a prime, let X_{sing(p)} denote the subcomplex consisting of points whose stabilizer has order divisible by p. For G a group and p a prime, let S_{p}(G) denote the poset of non-trivial finite p-subgroups of G.

Proposition 3.5. Suppose that X is a finite-dimensional contractible G-CW-complex with all stabilizers in 𝔓. For each prime p, the mod-p homology of X_{sing(p)} is isomorphic to the mod-p homology of the (realization of the) poset S_{p}(G).

Proof. Fix a prime p, and to simplify notation let S denote the realization of the poset S_{p}(G). For P a non-trivial p-subgroup of G, let X^{P} denote the points fixed by P, and let S_{≥P} denote the realization of the subposet of S_{p}(G) consisting of all p-subgroups that contain P. By the P. A. Smith theorem [3], each X^{P} is mod-p acyclic. Each S_{≥P} is contractible since it is equal to a cone with apex P. Let P and Q be p-subgroups of G, and let R = ⟨P, Q⟩, the subgroup of G generated by P and Q. If R is a p-group then X^{P} ∩ X^{Q} = X^{R}, and otherwise X^{P} ∩ X^{Q} is empty. Similarly, S_{≥P} ∩ S_{≥Q} = S_{≥R} if R is a p-group and S_{≥P} ∩ S_{≥Q} is empty if R is not a p-group.

Since each X^{P} is mod-p acyclic, the mod-p homology H_{∗}(X_{sing(p)}) is isomorphic to the mod-p homology of the nerve of the covering X_{sing(p)} = ⋃_{P} X^{P}. Similarly, the mod-p homology H_{∗}(S_{p}(G)) is isomorphic to the mod-p homology of the nerve of the covering S_{p}(G) = ⋃_{P} S_{≥P}. By the remarks in the first paragraph, these two nerves are isomorphic. □

Proposition 3.6. Let k be a finite field, and let G be the group of k-points of a reductive algebraic group over k, whose commutator subgroup has k-rank n. (For example, G = SL_{n+1}(k), or GL_{n+1}(k).) Any finite-dimensional contractible G-CW-complex with stabilizers in 𝔓 has dimension at least n.

Proof. The hypotheses on G imply that G acts on a spherical building Δ of dimension n − 1 [1, 5, Appendix on algebraic groups]. Any such building is homotopy equivalent to a wedge of (n − 1)-spheres. Quillen has shown that Δ is homotopy equivalent to the realization of S_{p}(G), where p is the characteristic of the field k [21, Proposition 2.1 and Theorem 3.1]. It follows that S_{p}(G) is homotopy equivalent to a wedge of (n − 1)-spheres, and in particular the mod-p homology group H_{n−1}(S_{p}(G)) is non-zero.

Now suppose that X is a finite-dimensional contractible G-CW-complex with stabilizers in 𝔓. Using Proposition 3.5, one sees that the mod-p homology group H_{n−1}(X_{sing(p)}) is non-zero. It follows that X must have dimension at least n. □

Remark 3.7. In [22] it is shown that if G is a finite simple group of Lie type, of Lie rank n, then any contractible G-CW-complex of dimension strictly less than n contains a point fixed by G. (Theorem 1 of [22] contains the additional hypothesis that the G-CW-complex should be finite, but this is not used in the proof.) A similar argument to that used in [22, Theorem 2] was used in [13] to show that when G = SL_{n+1}(𝔽_{p}), every contractible G-CW-complex without a global fixed point has dimension at least n. Note that Proposition 3.6 applies in greater generality than these results. For example, the Conner-Floyd construction [6] shows that whenever the multiplicative group of k does not have prime-power order, there is, for any n ≥ 1, a 4-dimensional contractible Gl_{n}(k)-CW-complex without a global fixed point.

Theorem 3.8. There are the following strict containments and equalities between classes of groups:

(i) 𝔉 ⊊ H_{1}𝔓; (ii) H_{1}𝔓 ⊊ H_{1}𝔉; (iii) H𝔉 = H𝔓.

Proof. Corollary 3.2 shows that 𝔉 ⊆ H_{1}𝔓. The free product of two cyclic groups of prime order is in H_{1}𝔓 and is not finite. The claim that H𝔉 = H𝔓 follows from the inequalities 𝔓 ⊆ 𝔉 ⊆ H_{1}𝔓, and the claim H_{1}𝔓 ⊆ H_{1}𝔉 follows from 𝔓 ⊆ 𝔉.

It remains to exhibit a group G that is in H_{1}𝔉 but not in H_{1}𝔓. Let G = SL_{∞}(𝔽_{p}), the direct limit of the groups G_{n} = SL_{n}(𝔽_{p}), where G_{n} is included in G_{n+1} as the ‘top corner’. As a countable locally-finite group, G acts with finite stabilizers on a tree. (Explicitly, the vertex set V and edge set E are both equal as G-sets to the disjoint union of the sets of cosets G/G_{1} ∪ G/G_{2} ∪ ⋯, with the edge gG_{i} joining the vertex gG_{i} to the vertex gG_{i+1}.) It follows that G ∈ H_{1}𝔉. By Proposition 3.6, G cannot be in H_{1}𝔓. □

Remark 3.9. Let G be a group in H𝔉 that is also of type FP_{∞}. By a result of Kropholler [10], there is a bound on the orders of finite subgroups of G, and Kropholler-Mislin show that G is in H_{1}𝔉 [11]. Corollary 3.2 shows that G is in H_{1}𝔓.

4. Cohomology relative to a class of groups

Let Δ denote a G-set, and let ℤΔ denote the corresponding Gmodule. For δ ∈ Δ, we write G_{δ} for the stabilizer of δ. A short exact sequence A ↣ B ↠ C of G-modules is said to be Δ-split if and only if it splits as a sequence of G_{δ}-modules for each δ ∈ Δ. Equivalently, the sequence is Δ-split if and only if the following sequence of ℤG-modules splits: A ⊗ ℤΔ ↣ B ⊗ ℤΔ ↠ C ⊗ ℤΔ [18].

We say a G module is Δ-projective if it is a direct summand of a G-module of the form N ⊗ ℤΔ, where N is an arbitrary G-module. Δ-projectives satisfy analogue properties to ordinary projectives. Furthermore, for each δ, and each G_{δ}-module M, the induced module Ind_{G_{δ}}^{G}M is Δ-projective. Given two G-sets Δ_{1} and Δ_{2} and a G-map Δ_{1} → Δ_{2} then Δ_{1}-projectives are Δ_{2}-projective and Δ_{2}-split sequences are Δ_{1}-split. For more detail the reader is referred to [18].

Now suppose that F is a class of groups closed under taking subgroups. We consider G-sets Δ satisfying the following condition, for all H ≤ G:

(∗)  Δ^{H} ≠ ∅ ⟺ H ∈ F.

There are G-maps between any two G-sets satisfying condition (∗), and so we may define an F-projective module to be a Δ-split module for any such Δ. Similarly, an F-split exact sequence of G-modules is defined to be a Δ-split sequence. If Δ satisfies (∗) and M is any G-module, the module M ⊗ ℤΔ is F-projective and admits an F-split surjection to M. This leads to a construction of homology relative to F. An F-projective resolution of a module M is an F-split exact sequence

⋯ → P_{n+1} → P_{n} → ⋯ → P_{0} → M → 0,

where all P_{i} are F-projective. Group cohomology relative to F, denoted FH^{∗}(G; N) can now be defined as the cohomology of the cochain complex Hom_{G}(P_{∗}, N), where P_{∗} is an F-projective resolution of ℤ.

We say that a module M is of type FFP_{n} if M admits an Fprojective resolution in which P_{i} is finitely generated for 0 ≤ i ≤ n. It has been shown that modules of type FFP_{n} are of type FP_{n} [18]. We will say that a group G is of type FFP_{n} if the trivial G-module ℤ is of type FFP_{n}.

We now specialize to the cases when F = 𝔉 and F = 𝔓.

Theorem 4.1. The following properties hold.

(i) A short exact sequence of G-modules is 𝔉-split if and only if it is 𝔓-split.

(ii) A G module is 𝔉-projective if and only if it is 𝔓-projective. (iii) 𝔉H^{∗}(G, −) ≅ 𝔓H^{∗}(G, −)

Proof: (i) It is obvious that any 𝔉-split sequence is 𝔓-split, and the converse follows from a standard averaging argument. Let H be an arbitrary finite subgroup of G. Then |H| = ∏_{i=1,...,n} p_{i}^{a_{i}} where p_{i} are distinct primes and 0 < a_{i} ∈ ℤ. For each i, let n_{i} be the index n_{i} = [H : P_{i}]. Now consider a 𝔓-split surjection A ↠^{π} B. Let σ_{i} be a P_{i}-splitting of π, and define a map s_{i} by summing σ_{i} over the cosets of P_{i}:

s_{i}(b) = ∑_{t∈H/P_{i}} tσ_{i}(t^{−1}b).

For each P_{i} we obtain a map s_{i} : B → A, such that π ∘ s_{i} = n_{i} × id_{B}. There exist m_{i} ∈ ℤ so that ∑_{i} m_{i}n_{i} = 1, and the map s = ∑_{i} m_{i}s_{i} is the required H-splitting.

(ii) It is obvious that a 𝔓-projective module is 𝔉-projective. Now let P be 𝔉-projective. We may take a 𝔓-split surjection M ↠ P with M a 𝔓-projective. By (i) this surjection is 𝔉-split, and hence split. Thus P is a direct summand of a 𝔓-projective and so is 𝔓-projective.

(iii) now follows directly from (i) and (ii). □

Proposition 4.2. A group G is of type 𝔉FP_{0} if and only if G has only finitely many conjugacy classes of subgroups of prime power order.

Proof: Suppose that G has only finitely many conjugacy classes of subgroups in 𝔓. Let I be a set of representatives for the conjugacy classes of 𝔓-subgroups and set

Δ_{0} = ⨆_{P ∈I} G/P.

This G-set satisfies condition (∗) for 𝔓 and therefore the surjection ℤΔ_{0} ↠ ℤ is 𝔉-split and also ℤΔ_{0} is finitely generated.

To prove the converse we consider an arbitrary 𝔉-split surjection P_{0} ↠ ℤ with P_{0} a finitely generated 𝔉-projective. As in [18, 6.1] we can show that P_{0} is a direct summand of a module ⨁_{δ∈Δ_{f}} Ind_{G_{δ}}^{G}P_{δ}, where Δ_{f} is a finite G-set, the G_{δ} are finite groups and P_{δ} are finitely generated G_{δ}-modules. Therefore we might assume from now on that P_{0} is of the above form. Since there is a G-map Δ_{f} → Δ, where Δ satisfies condition (∗) the 𝔉-split surjection P_{0} ↠^{ε} ℤ is also Δ_{f}-split [18]. Consider now the following commutative diagram:

[figure: P_{0}; ε; β; ℤ; ℤΔ_{f}; ε_{f}; α; ℤ]

That we can find such an α follows from the fact that ε is Δ_{f}-split, and β exists since P_{0} is Δ_{f}-projective being a direct sum of induced modules, induced from G_{δ}, (δ ∈ Δ_{f}) to G.

As a next step we’ll show that ε_{f} is 𝔉-split. Take an arbitrary finite subgroup H of G and show that ε_{f} splits when restricted to H. Since ε is split by s, say, when restricted to H we can define the required splitting by β ∘ s.

Now let P be an arbitrary p-subgroup of G. Since the module ℤ[G/P] is 𝔉-projective, there exists a G-map φ, such that the following diagram commutes:

[figure: ℤΔ_{f}; ε_{f}; ℤ; ℤ[G/P]; φ; ℤ]

The image φ(P) of the identity coset P is a point of ℤΔ fixed by the action of P. If H is any group and ℤΩ is any permutation module, then the H-fixed points are generated by the orbit sums H.ω. Hence P must stabilize some point of Δ_{f}, since otherwise we would have that p divides ε_{f}φ(P) = εα(P) = 1, a contradiction. It follows that P is a subgroup of G_{δ} for some δ ∈ Δ_{f}. □

Note that being of type 𝔉FP_{0} does not imply that there are finitely many conjugacy classes of finite subgroups. In fact, the authors have examples with infinitely many conjugacy classes of finite subgroups, see [14]. Nevertheless this gives rise to the following conjecture:

Conjecture 4.3. A group G is of type 𝔉FP_{∞} if and only if G is of type FP_{∞} and has finitely many conjugacy classes of p-subgroups.

It is shown in [18] that any G of type 𝔉FP_{∞} is of type FP_{∞}, which together with Proposition 4.2 proves one implication in the above conjecture.

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Department of Mathematics, The Ohio State University, 231 W 18th Ave, Columbus Ohio 43210

Heilbronn Institute, University of Bristol, Royal Fort Annexe, Bristol BS8 1TW

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

School of Mathematics, University of Southampton, Southampton SO17 1BJ, UK

  E-mail address: B.E.A.Nucinkis@soton.ac.uk