“The type of the classifying space for a family of subgroups” by Wolfgang Lück

Key words: Classifying space for a family of subgroups, finiteness-conditions

AMS-classification number: 55R35

Abstract: The classifying space E(Γ, F) for a family F of subgroups of a group Γ is defined up to Γ-homotopy as a Γ-CW -complex E(Γ, F) such that E(Γ, F)^{H} is contractible if H belongs to F and is empty otherwise. We investigate the question whether there is a finite-dimensional Γ-CW -model, a finite Γ-CW -model or a Γ-CW -model of finite type for E(Γ, F) focusing on the case where F is the family of finite subgroups.

0. Introduction

Let Γ be a group and F be a family of subgroups, i.e. a set of subgroups of Γ which is closed under conjugation and taking subgroups.

Definition 0.1. A classifying space E(Γ, F) for F is a Γ-CW -complex such that E(Γ, F)^{H} is contractible for H ∈ F and empty otherwise.

If F is the family FIN of finite subgroups, we abbreviate

\underline{E}Γ = E(Γ, FIN).

The existence of E(Γ, F) and proofs that for any Γ-CW -complex X whose isotropy groups belong to F there is precisely one Γ-map up to Γ-homotopy from X to E(Γ, F) and thus that two such classifying spaces are Γ-homotopy equivalent, are presented in [9],[10, I.6]. A functorial “bar-type” construction is given in [8, section 7]. These classifying spaces occur in the Isomorphism-Conjectures in algebraic K- and L-theory [12, 1.6 on page 257] for F the family of virtually cyclic groups and in the Baum-Connes-Conjecture [2, Conjecture 3.15 on page 254] for F the family FIN of finite subgroups. The space \underline{E}Γ plays also a role in the extension of the Atiyah-Segal-Completion Theorem from finite to infinite groups [17]. Sometimes \underline{E}Γ is also called the classifying space for proper actions. Notice that E(Γ, F) for F the family consisting of one element, namely the trivial subgroup, is the same as the total space EΓ of the universal principal Γ-bundle EΓ ⟶ BΓ. If Γ is torsionfree, then \underline{E}Γ is the same as EΓ. If Γ is finite, then the one-point-space is a model for \underline{E}Γ.

In this paper we are dealing with the question whether one can find a d-dimensional Γ-CW -model, a finite Γ-CW -model or a Γ-CW -model of finite type for \underline{E}Γ. Recall that a Γ-CW -complex X is finite if and only if it consists of finitely many Γ-equivariant cells, or, equivalently, Γ\X is compact. It is called of finite type if each skeleton is finite. More information about Γ-CW -complexes can be found in [10, II.1 and II.2] and [15, section 1 and 2]. A survey of groups for which nice geometric Γ-CW -models for \underline{E}Γ exist can be found in [2, Section 2]. These include for instance (i) word-hyperbolic groups Γ for which the Rips complex yields a finite Γ-CW -model for \underline{E}Γ, (ii) discrete subgroups Γ ⊂ G of a Lie group G with finitely many path components for which G/K with the left Γ-action for a maximal compact subgroup K ⊂ G is a Γ-CW -model for \underline{E}Γ and (iii) groups Γ acting cellularly (without inversion) on trees with finite isotropy groups.

In Section 1 we give necessary conditions for the existence of a Γ-CW -model of \underline{E}Γ with prescribed dimensions of the H-fixed point sets \underline{E}Γ^{H} in terms of the Borel cohomology of the posets of non-trivial finite subgroups of the Weyl groups WH of the finite subgroups H ⊂ Γ. (see Theorem 1.6). The length l(H) ∈ {0, 1, … } of a finite group H is the supremum over all p for which there is a nested sequence H_{0} ⊂ H_{1} ⊂ … ⊂ H_{p} of subgroups H_{i} of H with H_{i} ≠ H_{i+1}. We do not know whether the necessary conditions above are also sufficient in general. However, if there is an upper bound l on the length l(H) of finite subgroups H of Γ, then the necessary conditions above are also sufficient (see Theorem 1.6).

We also give a necessary algebraic condition B(d) for a non-negative integer d for the existence of a d-dimensional Γ-CW -model for \underline{E}Γ, namely that for each finite subgroup H ⊂ Γ a ℤWH-module M whose restriction to ℤK is projective for any finite subgroup K ⊂ WH has a d-dimensional projective ℤWH-resolution (Notation 1.4 and Lemma 1.5). If there is an upper bound l on the length l(H) of finite subgroups H of Γ and B(d) is satisfied we will prove that there is a (max{3, d} + l(d + 1))-dimensional Γ-CW -model for \underline{E}Γ (Theorem 1.10). Such a result has already been proven for certain classes of groups by Kropholler and Mislin [13, Theorem B], for an in l exponential dimension estimate.

In Section 2 we show that \underline{E}Γ has a dm-dimensional Γ-CW -model if there is a subgroup Δ ⊂ Γ of finite index d with m-dimensional Δ-CW -model for \underline{E}Δ (Theorem 2.4).

In Section 3 we show for an exact sequence 1 → Δ → Γ → π → 1 such that there is an upper bound on the orders of finite subgroups of π that \underline{E}Γ has a finite-dimensional Γ-CW -model if \underline{E}Δ and \underline{E}π respectively have a finite-dimensional Δ-CW -model and a finitedimensional π-CW -model respectively (Theorem 3.1). We discuss to which extend a statement like this holds if we ask for finite models or models of finite type (Theorem 3.2 and Theorem 3.3).

In Section 4 we prove that \underline{E}Γ has a Γ-CW -model of finite type if and only if there is a CW -model of finite type for BWH for all finite subgroups H ⊂ Γ and Γ contains only finitely many conjugacy classes of finite subgroups. (Theorem 4.2).

In Section 5 we deal with the question whether there is a finitely dominated or finite Γ-CW -complex model for \underline{E}Γ (Theorem 5.1 and Remark 5.2).

In Section 6 we consider the special case of groups of finite virtual cohomological dimension. Provided that Γ contains a torsionfree group of finite index, Γ satisfies B(d) if and only if Γ has virtual cohomological dimension ≤ d (Theorem 6.3). If l is an upper bound on the length l(H) of finite subgroups H of Γ and Γ has virtual cohomological dimension ≤ d, then we will prove that there is a (max\{3, d\} + l)-dimensional Γ-CW -model for \underline{E}Γ (Theorem 6.4).

Finally we discuss some open problems in Section 7.

We will always work in the category of compactly generated spaces (see [20] and [22, I.4]). A Γ-space or a ℤΓ-module respectively is always to be understood as a left Γ-space or left ℤΓ-module respectively. The letter Γ stands always for a discrete group. The paper is organized as follows:

0. Introduction
1 . Finite-dimensional classifying spaces
2. Coinduction
3. Short exact sequences
4. Classifying spaces of finite type
5. Finitely dominated and finite classifying spaces
6. Groups with finite virtual cohomological dimension
7. Problems

References

The author wants to thank Frank Connolly, Peter Kropholler and Guido Mislin for stimulating and fruitful discussions and the referee for its careful report and providing an answer to a problem which was stated in an earlier version.

1. Finite-dimensional classifying spaces

In this section we deal with the question whether there are finite-dimensional Γ-CW - models for \underline{E}Γ.

Define the Γ-poset

P(Γ) := {K | K ⊂ Γ finite, K ≠ 1}.  (1.1)

An element γ ∈ Γ sends K to γKγ^{−1} and the poset-structure comes from inclusion of subgroups. Denote by |P(Γ)| the geometric realization of the category given by the poset P(Γ). This is a Γ-CW -complex but in general not proper, i.e. it can have points with infinite isotropy groups.

Let NH be the normalizer and let WH := NH/H be the Weyl group of H ⊂ Γ. Notice for a Γ-space X that X^{H} inherits a WH-action. Denote by σ_{Γ}(X) the singular set of the Γ-space X, i.e. the set of points with non-trivial isotropy groups. Notice that σ_{Γ′}(X) may differ from σ_{Γ}(X) for a subgroup Γ′ ⊂ Γ. Denote by CX the cone over X. Notice that C∅ is the one-point-space. The meaning of |P(Γ)| lies in the following result which follows from [7, Lemma 2.4].

Lemma 1.2. There is a Γ-equivariant map

f : (\underline{E}Γ, σ_{Γ}(\underline{E}Γ)) ⟶ (C|P(Γ)|, |P(Γ)|),

which is a (non-equivariant) homotopy equivalence.

If H and K are subgroups of Γ and H is finite, then Γ/K^{H} is a finite union of WH-orbits of the shape WH/L for finite L ⊂ WH. Now one easily checks

Lemma 1.3. The WH-space \underline{E}Γ^{H} is a WH-CW -model for \underline{E}WH. In particular, if \underline{E}Γ has a Γ-CW -model which is finite, of finite type or d-dimensional respectively, then there is a WH-model for \underline{E}WH which is finite, of finite type or d-dimensional respectively.

Notation 1.4. Let d ≥ 0 be an integer. A group Γ satisfies the condition b(d) or b(< ∞) respectively if any ℤΓ-module M with the property that M restricted to ℤK is projective for all finite subgroups K ⊂ Γ has a projective ℤΓ-resolution of dimension d or of finite dimension respectively. A group Γ satisfies the condition B(d) if WH satisfies the condition b(d) for any finite subgroup H ⊂ Γ.

The length l(H) ∈ {0, 1, … } of a finite group H is the supremum over all p for which there is a nested sequence H_{0} ⊂ H_{1} ⊂ … ⊂ H_{p} of subgroups H_{i} of H with H_{i} ≠ H_{i+1}.

Lemma 1.5. Suppose that there is a d-dimensional Γ-CW -complex X with finite isotropy groups such that H_{p}(X; ℤ) = H_{p}(∗, ℤ) for all p ≥ 0 holds. This assumption is for instance satisfied if there is a d-dimensional Γ-CW -model for \underline{E}Γ. Then Γ satisfies condition B(d).

Proof. Let H ⊂ Γ be finite. Then X/H satisfies H_{p}(X/H; ℤ) = H_{p}(∗, ℤ) for all p ≥ 0 [4, III.5.4 on page 131]. Let C_{∗} be the cellular ℤWH-chain complex of X/H. This is a d-dimensional resolution of the trivial ℤWH-module ℤ and each chain module is a sum of ℤWH-modules of the shape ℤ[WH/K] for some finite subgroup K ⊂ WH. Let N be a ℤWH-module such that N is projective over ℤK for any finite subgroup K ⊂ WH. Then C_{∗} ⊗_{ℤ} N with the diagonal WH-operation is a d-dimensional projective ℤWH-resolution of N.

Theorem 1.6. Let Γ be a group. Suppose that we have for any finite subgroup H ⊂ Γ an integer d(H) ≥ 3 such that d(H) ≥ d(K) for H ⊂ K and d(H) = d(K) if H and K are conjugated in Γ. Consider the following statements:

1. There is a Γ-CW -model \underline{E}Γ such that for any finite subgroup H ⊂ Γ

dim(\underline{E}Γ^{H}) = d(H).

2. We have for any finite subgroup H ⊂ Γ and for any ℤWH-module M

H_{ℤWH}^{d(H)+1}(EWH × (C|P(WH)|, |P(WH)|); M) = 0.

3. We have for any finite subgroup H ⊂ Γ that its Weyl group WH satisfies b(<∞) and that there is a subgroup Δ(H) ⊂ WH of finite index such that for any ℤΔ(H)-module M

H_{ℤΔ(H)}^{d(H)+1}(EΔ(H) × (C|P(WH)|, |P(WH)|); M) = 0.

Then (1) implies both (2) and (3). If there is an upper bound on the length l(H) of the finite subgroups H of Γ, then these statements (1), (2) and (3) are equivalent.

In the case that Γ has finite virtual cohomological dimension a similar result is proven in [7, Theorem III].

Example 1.7. Suppose that Γ is torsionfree. Then Theorem 1.6 reduces to the well-known result [6, Theorem VIII.3.1 on page 190,Theorem VIII.7.1 on page 205] that the following assertions are equivalent for an integer d ≥ 3:

1. There is a d-dimensional CW -model for BΓ;

2. Γ has cohomological dimension ≤ d;

3. Γ has virtual cohomological dimension ≤ d.

Example 1.8. In this example we use the notation of Theorem 1.6. If WH is torsionfree, then

H_{ℤWH}^{d(H)+1}(EWH × (C|P(WH)|, |P(WH)|); M) = H_{ℤWH}^{d(H)+1}(EWH; M);
H_{ℤΔ(H)}^{d(H)+1}(EΔ(H) × (C|P(WH)|, |P(WH)|); M) = H_{ℤΔ(H)}^{d(H)+1}(EΔ(H); M),

and the condition that H_{ℤWH}^{d(H)+1}(EWH×(C|P(WH)|, |P(WH)|); M) = 0 or H_{ℤΔ(H)}^{d(H)+1}(EΔ(H)× (C|P(WH)|, |P(WH)|); M) = 0 respectively holds for all ℤWH-modules M or ℤΔ(H)- modules M respectively is equivalent to the existence of a d(H)-dimensional model for BWH or BΔ(H) respectively.

If WH contains a non-trivial normal finite subgroup L, then |P(WH)| is contractible and

H_{ℤWH}^{d(H)+1}(EWH × (C|P(WH)|, |P(WH)|); M) = 0;
H_{ℤΔ(H)}^{d(H)+1}(EΔ(H) × (C|P(WH)|, |P(WH)|); M) = 0.

Namely, define maps of posets C, F : P(WH) ⟶ P(WH) by C(K) = L and F (K) = ⟨K, L⟩ where the subgroup ⟨K, L⟩ generated by K and L is finite since L is normal in WH. Since K ⊂ F (K) and L ⊂ F (K) holds for all finite subgroups K, there are natural transformations between the functors id and F and the functors C and F of the category given by the poset P(WH) and hence the maps induced by these functors on the geometric realizations are homotopic. Hence the identity is homotopic to the constant map which is the map induced by C.

The proof of Theorem 1.6 needs some preparation. The next lemma deals with the question whether a Γ-CW -complex can be made contractible by attaching free cells of bounded dimension and is the key ingredient in the induction step in the proof of Theorem 1.6.

Lemma 1.9. Let X be a Γ-CW -complex and d ≥ 3 be an integer. Consider the following statements.

1. There is a Γ-CW -complex Y which is obtained from X by attaching free cells Γ × D^{n} of dimension n ≤ d and is contractible.

2. There is a Γ-CW -complex Y which is obtained from EΓ × X by attaching free cells Γ × D^{n} of dimension n ≤ d and is contractible.

3. H_{n}(X; ℤ) = 0 for n ≥ d and for any (d − 2)-connected Γ-CW -complex Z which is obtained from X by attaching free cells Γ × D^{n} of dimension n ≤ d − 1 the ℤΓ-module H_{d−1}(Z; ℤ) is ℤΓ-projective.

4. H_{n}(X; ℤ) = 0 for n ≥ d and there is a (d − 2)-connected Γ-CW -complex Z which is obtained from X by attaching free cells Γ × D^{n} of dimension n ≤ d − 1 such that the ℤΓ-module H_{d−1}(Z; ℤ) is ℤΓ-projective.

5. The ℤΓ-chain complex C_{∗}(EΓ × (CX, X)) is ℤΓ-chain homotopy equivalent to a ddimensional projective ℤΓ-chain complex.

6. H_{n}(X; ℤ) = 0 for n ≥ d + 1 and H_{ℤΓ}^{d+1}(EΓ × (CX, X); M) = 0 for any ℤΓ-module M.

7. H_{n}(X; ℤ) = 0 for n ≥ d+1, the group Γ satisfies b(<∞) and there is a subgroup Γ_{0} ⊂ Γ of finite index such that H_{ℤΓ_{0}}^{d+1}(EΓ_{0} × (CX, X); M) = 0 holds for any ℤΓ_{0}-module M.

8. There is a non-negative integer e such that H_{n}(X; ℤ) = 0 for n ≥ d − e and the group Γ satisfies b(e).

Then we have

(1) ⟺ (2) ⟺ (3) ⟺ (4) ⟺ (5) ⟺ (6).

If X is finite-dimensional and all isotropy groups of X are finite then (1) ⟹ (7).

If X^{H} is contractible for all non-trivial finite subgroups H ⊂ Γ, then (7) ⟹ (1) and (8) ⟹ (1).

Proof. (1) ⟹(2) Let Z ⊃ X be the given extension. Then one constructs a pair of Γ-CW - complexes (Z′, EΓ×X) such that Z′ is obtained from EΓ×X by attaching free cells Γ×D^{n} of dimension n ≤ d and for the Γ-space Z′′ defined by the Γ-pushout

EΓ × X ⟶ Z′
pr↓  \sqrt{pr}
X ⟶ Z′′

there is a Γ-homotopy equivalence of pairs relative X from (Z′′, X) to (Z, X). Then Z′ is contractible and the desired Γ-CW -complex.

In fact there is a bijective correspondence between the cells Γ × D^{n} of Z′ − (EΓ × X) and of Z − X for all n. The construction is based on the observation that for any Γ-map f : A ⟶ B of Γ-spaces which is a (non-equivariant) homotopy equivalence any Γ-map Γ × S^{n} ⟶ B lifts up to Γ-homotopy to a Γ-map Γ × S^{n} ⟶ A and changing the Γhomotopy type of the attaching maps does not change the Γ-homotopy type of a Γ-CW - complex. Details can be found in [15, page 288 - 289].

(2) ⟹ (5) There is a Γ-homotopy equivalence (f, id) : (Y, EΓ×X) ⟶ EΓ×(CX, X) since both EΓ × CX and Y are contractible.

(5) ⟹(3) Define a Γ-CW -complex Y by the Γ-pushout

EΓ × X ⟶ EΓ × CX
pr↓  \sqrt{pr}
X ⟶  Y

for pr : EΓ × X ⟶ X the projection. Since the inclusion of EΓ × X into EΓ × CX is a cofibration and pr is a (non-equivariant) homotopy equivalence, \overline{pr} is a (non-equivariant) homotopy equivalence. Hence Y is contractible. Obviously C_{∗}(Y, X) and C_{∗}(EΓ×(CX, X)) are ℤΓ-chain isomorphic. Choose a Γ-map f : Z ⟶ Y which induces the identity on X and a ℤΓ-chain homotopy equivalence g_{∗} : C_{∗}(Y, X) ⟶ P_{∗} for a d-dimensional projective ℤΓ-chain complex P_{∗}. Let C_{∗} be the mapping cone of the ℤΓ-chain map g_{∗} ∘ C_{∗}(f, id) : C_{∗}(Z, X) ⟶ P_{∗}. Since the relative dimension of (Z, X) is d − 1 and the dimension of P_{∗} is d, C_{∗} is a d-dimensional projective ℤΓ-chain complex. Since Z is (d − 2)-connected and Y is contractible, H_{j}(C_{∗}) is trivial for j ≤ d − 1 and is H_{d−1}(Z; ℤ) for j = d. Hence we obtain an exact sequence of ℤΓ-modules

0 ⟶ H_{d−1}(Z; ℤ) ⟶ C_{d} ⟶ C_{d−1} ⟶ … ⟶ C_{0} ⟶ 0.

This implies that H_{d−1}(Z; ℤ) is a projective ℤΓ-module. We conclude H_{j}(X; ℤ) = 0 for j ≥ d from the long homology sequence associated to EΓ × (CX, X) and the fact that H_{j}(EΓ × (CX, X); ℤ) = 0 for j ≥ d + 1.

(3) ⟹ (4) is obvious.

(4) ⟹(1) By the Eilenberg swindle there exists a free ℤΓ-module F such that F ⊕H_{d−1}(Z) is free. Hence we can achieve by attaching free (d − 1)-dimensional cells Γ × D^{d−1} with trivial attaching maps to Z that H_{d−1}(Z) is free. The Hurewicz homomorphism π_{d−1}(Z) ⟶ H_{d−1}(Z; ℤ) is bijective since Z is (d − 2)-connected and d ≥ 3. Now choose a ℤΓ-basis for π_{d−1}(Z) and attach for each basis element which is given by a map g : S^{d−1} ⟶ Z a free Γ-cell Γ × D^{d} to Z by the attaching map Γ × S^{d−1} ⟶ Z (γ, s) ↦ γ · g(s). The resulting space Y is obtained from X by attaching free Γ-cells Γ × D^{n} of dimension n ≤ d. It is (d − 2)-connected and in particular simply-connected by construction. By inspecting the long homology sequence of (Y, Z) one concludes that the homology H_{j}(Y; ℤ) is trivial for 1 ≤ j ≤ d − 1 and the map induced by the inclusion H_{d}(Z; ℤ) ⟶ H_{d}(Y; ℤ) is surjective. Using the long exact sequences of the pair (Z, X) one concludes that H_{d}(Y; ℤ) is trivial. Since H_{j}(Y; ℤ) = H_{j}(X; ℤ) = 0 holds for j ≥ d + 1, the space Y must be contractible.

(5) ⟺ (6) If we would substitute in (6) the condition H_{n}(X; ℤ) = 0 for n ≥ d + 1 by the sharper condition H_{n}(X; ℤ) = 0 for n ≥ d, (5) ⟺ (6) would follow from [15, Proposition 11.10 on page 221] or [21]. Hence it remains to show that H_{d}(X; ℤ) = 0 holds provided that H_{ℤΓ}^{d+1}(EΓ × (CX, X); M) = 0 for any ℤΓ-module M. Let A be any abelian group. Equip i_{♯}A := hom_{ℤ}(ℤΓ, A) with the left ℤΓ-module structure given by (γ · f)(x) = f(x · γ) for γ ∈ Γ, f ∈ i_{♯}A and x ∈ ℤΓ. This is the coinduction of A with respect to the inclusion of the trivial group in Γ. Since coinduction is the right adjoint of restriction, we get

H^{d}(X; A) ≅ H_{ℤ}^{d+1}(EΓ × (CX, X); A)
≅ H_{ℤΓ}^{d+1}(EΓ × (CX, X); i_{♯}A)
≅ 0.

From the universal coefficient theorem we conclude hom_{ℤ}(H_{d}(X; ℤ), A) = 0 for any abelian group A. This implies H_{d}(X; ℤ) = 0.

Now we show (1) ⟹ (7) provided that X is finite-dimensional and the isotropy groups of X are finite. If i : Γ_{0} ⟶ Γ is the inclusion and i_{♯} denotes coinduction with i, we have

H_{ℤΓ_{0}}^{d+1}(EΓ_{0} × (CX, X); M) = H_{ℤΓ}^{d+1}(EΓ × (CX, X); i_{♯}M)

for any ℤΓ_{0}-module M. Since we already know (1) ⟹ (6), it suffices to show that Γ satisfies b(d) if (1) holds. Then C_{∗}(Y) is a finite-dimensional ℤΓ-resolution of the trivial ℤΓ-module ℤ and each C_{n}(Y) is a direct sum of ℤΓ-modules of the shape ℤΓ/L for appropriate finite subgroups L ⊂ Γ. If N is a ℤΓ-module which is projective over ℤH for all finite subgroups H ⊂ Γ, then C_{∗}(X) ⊗_{ℤ} N with the diagonal Γ-action is a finite-dimensional projective ℤΓ-resolution of N.

From now on suppose that X^{K} is contractible for any non-trivial finite subgroup K ⊂ Γ. Notice that then for any non-trivial finite subgroup H ⊂ Γ the singular set σ_{H}(X) of the H-space X is contractible because of [7, Lemma 2.5 on page 22] because σ_{H}(X) is the union of the sets X^{K} for K ⊂ H with K ≠ 1 and X^{K_{1}} ∩ X^{K_{2}} = X^{⟨K_{1},K_{2}⟩}, where ⟨K_{1}, K_{2}⟩ is the subgroup of H generated by K_{1} and K_{2}.

(7) ⟹ (3) If the Γ-space X satisfies (7), the Γ_{0}-space X satisfies (6) and hence (3). Therefore H_{n}(X; ℤ) = 0 for n ≥ d and H_{d−1}(Z; ℤ) is projective over ℤΓ_{0}. Hence it remains to show that H_{d−1}(Z; ℤ) is projective over ℤΓ. Because of [7, Lemma 4.1 (a) on page 26] it suffices to show that the cohomological dimension of the ℤΓ-module H_{d−1}(Z; ℤ) is finite. As Γ satisfies b(<∞) by assumption it suffices to prove for any finite subgroup H ⊂ Γ that the ℤH-module H_{d−1}(Z; ℤ) is projective. Since the singular set σ_{H}(X) of the H-space X is contractible and agrees with the singular set σ_{H}(Z) of the H-space Z we conclude H_{j}(Z; ℤ) = H_{j}(Z, σ_{H}(Z); ℤ) for j ≥ 1. Since Z is (d − 2)-connected and the relative dimension of (Z, σ_{H}(Z)) is less or equal to d − 1, we obtain an exact sequence of ℤH-modules

0 ⟶ H_{d−1}(Z; ℤ) ⟶ C_{d−1}(Z, σ_{H}(Z)) ⟶ … ⟶ C_{0}(Z, σ_{H}(Z)) ⟶ 0.

Since each C_{j}(Z, σ_{H}(Z)) is ℤH-free, the ℤH-module H_{d−1}(Z; ℤ) is projective.

(8) ⟹ (4) Assume that the Γ-CW -complex Z′ is obtained from X by attaching free cells Γ × D^{n} of dimension n ≤ d − e − 1 and is (d − e − 2)-connected. By the argument above H_{d−e−1}(Z′; ℤ) is ℤH-projective for any finite subgroup H ⊂ Γ. Since Γ satisfies b(e) by assumption, the cohomological dimension of the ℤΓ-module H_{d−e−1}(Z′; ℤ) is less or equal to e. Now attach free cells Γ × D^{n} of dimension d − e ≤ n ≤ d − 1 to Z′ such that the resulting Γ-CW -complex Z is (d−2)-connected. We conclude from the long exact homology sequences associated to (Z, Z′), (Z, X) and (Z′, X) that H_{j}(Z, Z′; ℤ) = 0 for d − e + 1 ≤ j ≤ d − 2, H_{d−1}(Z; ℤ) = H_{d−1}(Z, Z′; ℤ) and H_{d−e−1}(Z′; ℤ) = H_{d−e}(Z, Z′; ℤ) hold. Hence we obtain an exact sequence

0 ⟶ H_{d−1}(Z; ℤ) ⟶ C_{d−1}(Z, Z′) ⟶ … ⟶ C_{d−e}(Z, Z′) ⟶ H_{d−e−1}(Z′; ℤ) ⟶ 0

This implies that H_{d−1}(Z; ℤ) is a projective ℤΓ-module. This finishes the proof of Lemma 1.9.

Proof. Now we are ready to give the proof of Theorem 1.6. First we show that (1) implies both (2) and (3). This follows directly from Lemma 1.2, Lemma 1.3 and the implication (1) ⟹ (6) and (1) ⟹ (7) of Lemma 1.9 applied to the singular set X = σ_{WH}(\underline{E}WH) of the WH-space \underline{E}WH. It remains to show that (1) holds provided that there is an upper bound l on the length l(H) of finite subgroups H ⊂ Γ and (2) or (3) is true.

We construct inductively a nested sequence of Γ-CW -complexes

∅ = X[l + 1] ⊂ X[l] ⊂ … ⊂ X[0]

with the following properties. The Γ-CW -complex X[n] is obtained from X[n + 1] by attaching cells of the type Γ/H × D^{k} for finite subgroups H ⊂ Γ with l(H) = n and k ≤ d(H) and X[n]^{H} is contractible for any finite subgroup H ⊂ Γ with l(H) ≥ n. Then X[0] is the desired model for \underline{E}Γ. The induction begin n = l + 1 is obvious, the induction step from n + 1 to n done as follows.

Let H ⊂ Γ be a finite subgroup with l(H) = n. Since d(K) ≤ d(H) holds for all finite subgroups K ⊂ Γ with H ⊂ K, the dimension of X[n + 1]^{H} is less or equal to d(H). In particular H_{n}(X[n + 1]^{H}; ℤ) = 0 for n ≥ d(H) + 1. Fix any ℤWH-module M. Let f : X[n + 1] ⟶ \underline{E}Γ be a Γ-map. It induces a homotopy equivalence f^{K} for all finite subgroups K ⊂ Γ with l(K) ≥ n + 1. Hence the WH-map σ_{WH}(f^{H}) : σ_{WH}(X[n + 1]^{H}) ⟶ σ_{WH}(\underline{E}Γ^{H}) induced by f is a (non-equivariant) homotopy equivalence by [7, Lemma 2.5 on page 22] since σ_{WH}(X[n + 1]^{H}) is the union of the spaces X[n + 1]^{K} for all finite subgroup K ⊂ Γ with H ⊂ K, H ≠ K ∩ NH and similiar for σ_{WH}(\underline{E}Γ^{H}) and any such group K satisfies l(K) ≥ n + 1. We get from Lemma 1.2 and Lemma 1.3 an isomorphism

H_{ℤWH}^{d(H)+1}(EWH × (Cσ_{WH}(X[n + 1]^{H}), σ_{WH}(X[n + 1]^{H})); M)
⟶^{≅} H_{ℤWH}^{d(H)+1}(EWH × (Cσ_{WH}(\underline{E}Γ^{H}), σ_{WH}(\underline{E}Γ^{H})); M)
⟶^{≅} H_{ℤWH}^{d(H)+1}(EWH × (Cσ_{WH}(\underline{E}WH), σ_{WH}(\underline{E}WH)); M)
⟶^{≅} H_{ℤWH}^{d(H)+1}(EWH × (C|P(WH)|, |P(WH)|); M)
= 0.

Since X[n + 1]^{H} is obtained from σ_{WH}(X[n + 1]^{H}) by attaching free WH-cells WH × D^{k} for k ≤ d(H), we get for any ℤWH-module M an isomorphism

H_{ℤWH}^{d(H)+1}(EWH × (CX[n + 1]^{H}, X[n + 1]^{H}); M)
⟶^{≅} H_{ℤWH}^{d(H)+1}(EWH × (Cσ_{WH}(X[n + 1]^{H}), σ_{WH}(X[n + 1]^{H})); M) = 0.

Similiarly we get H_{ℤΔ(H)}^{d(H)+1}(EΔ(H)×(CX[n+1]^{H}, X[n+1]^{H}); M) = 0 for any ℤΔ(H)-module M. Now the implications (6) ⟹ (1) and (7) ⟹ (1) of Lemma 1.9 yield the existence of a contractible WH-CW -complex Y (H) which is obtained from X[n + 1]^{H} by attaching free cells WH × D^{m} of dimension m ≤ d(H). Now define X[n] as the Γ-push out

∐_{(H)} Γ ×_{NH} X[n + 1]^{H} ⟶^{i} ∐_{(H)} Γ ×_{NH} Y (H)
p↓  ↓
X[n + 1]  ⟶  X[n]

where (H) runs through the conjugacy classes of finite subgroups H ⊂ Γ with l(H) = n, i is the obvious inclusion and p sends (γ, x) ∈ Γ ×_{NH} X[n + 1]^{H} to γx ∈ X[n + 1]. Since X[n]^{H} = Y (H) and X[n]^{H} = X[n + 1]^{H} respectively for any finite subgroup H ⊂ Γ with l(H) = n and l(H) > n respectively, one easily verifies that X[n] has the desired properties. This finishes the proof of Theorem 1.6.

The next result has already been proven by Kropholler and Mislin [13, Theorem B] for certain classes of groups and in l exponential dimension estimate (see also Remark 1.12). We will give an in l-linear one.

Theorem 1.10. Let Γ be a group and let l ≥ 0 and d ≥ 0 be integers such that the length l(H) of any finite subgroup H ⊂ Γ is bounded by l and Γ satisfies B(d). Then there is a Γ-CW -model for \underline{E}Γ such that for any finite subgroup H ⊂ Γ

dim(\underline{E}Γ^{H}) = max\{3, d\} + (l − l(H))(d + 1)

holds. In particular \underline{E}Γ has dimension max\{3, d\} + l(d + 1).

Proof. The proof is a variation of the proof of Theorem 1.6 now using in the induction step the implication (8) ⟹ (1) proven in Lemma 1.9 instead of the implications (6) ⟹ (1) and (7) ⟹ (1).

Remark 1.11. There are groups for which there is a 1-dimensional Γ-CW -model for \underline{E}Γ but no upper bound l on the length l(H) of finite subgroups. Namely, any group whose rational cohomological dimension is less or equal to 1 admits a 1-dimensional model for \underline{E}Γ [11]. In particular any countable locally finite group has a 1-dimensional model for \underline{E}Γ.

Remark 1.12. In [13] the class HFIN of hierarchically decomposable groups and the ℤΓmodule B(Γ, ℤ) of bounded functions from Γ to ℤ is investigated and related to the question of the existence of a finite-dimensional Γ-CW -model for \underline{E}Γ. By definition HFIN is the smallest class of group which contains all finite groups and which contains Γ if there is a finite-dimensional contractible Γ-CW -complex whose isotropy groups belong to the class. Define the following classes of groups where hdim means the homological dimension:

A := {Γ | hdim_{ℤΓ}(B(Γ, ℤ)) < ∞};
B := {Γ | Γ satisfies b(d)};
C := {Γ | Γ satisfies B(d)};
D := {Γ | there is a finite-dimensional Γ-CW -complex X with finite isotropy groups
and H_{p}(X; ℤ) = H_{p}(∗, ℤ) for p ≥ 0};
E := \{Γ | there is a finite-dimensional Γ-CW -model for \underline{E}Γ\};
L := {Γ | there is an upper bound l on the length l(H) of finite subgroups H ⊂ Γ}.

Then

A ∩ HFIN = B ∩ HFIN = C ∩ HFIN

follows directly from [13, Proposition 6.1, Lemma 7.3] and the fact that B(Γ, ℤ) is projective over ℤH for any finite subgroup H ⊂ Γ [14]. We conclude

A ∩ HFIN ∩ L = B ∩ HFIN ∩ L = C ∩ L = D ∩ L = E ∩ L

from Lemma 1.5 and Theorem 1.10.

2. Coinduction

In this section we study coinduction and what it does to a classifying space for a family.

Definition 2.1. Let i : Δ ⊂ Γ be an inclusion of groups. Define the coinduction with i of a Δ-space X to be the Γ-space

i_{♯}X := map_{Δ}(Γ, X),

where map_{Δ}(Γ, X) is the space of Δ-equivariant maps of Δ-spaces. The Γ-action is given by (γf)(γ_{0}) := f(γ_{0}γ) for f : Γ → X and γ, γ_{0} ∈ Γ. Given a family F of subgroups of Δ, define the family i_{♯}F of subgroups of Γ by the set of subgroups H ⊂ Γ for which γ^{−1}Hγ ∩ Δ belongs to F for all γ ∈ Γ.

Notice that i_{♯} is the right adjoint of the restriction functor which sends a Γ-space Y to the Δ-space i^{∗}Y, namely, there is a natural homeomorphism

map_{Δ}(i^{∗}Y, X) ⟶^{≅} map_{Γ}(Y, i_{♯}X).  (2.2)

The next result reduces for torsionfree Γ to Serre’s Theorem that there is a finitedimensional model for BΓ if Γ contains a subgroup Δ of finite index with finite-dimensional BΔ ([6, Theorem VIII.3.1. on page 190, Corollary VIII.7.2 on page 205], [18]).

Theorem 2.3. Let i : Δ ⊂ Γ be an inclusion of groups and let F be a family of subgroups of Δ. Then

1. If X is a Δ-CW -complex of dimension d and the index [Γ : Δ] is finite, then i_{♯}X is Γ-homotopy equivalent to a d · [Γ : Δ]-dimensional Γ-CW -complex;

2. i_{♯}E(Δ, F)^{H} is empty for H ∉ i_{♯}F and is contractible for H ∈ i_{♯}F.

Proof. 1.) Let l be the index [Γ : Δ]. Fix elements γ_{1}, γ_{2}, …, γ_{l} in Γ such that Δ\Γ = {Δγ_{k} | k = 1, 2, …, l}. Evaluation at γ_{k} defines a homeomorphism

ev : i_{♯}X ⟶^{≅} ∏_{k=1}^{l} X.

Let G be the semi-direct product of Δ^{l} = ∏_{k=1}^{l} Δ and the symmetric group S_{l} in l-letters with respect to the action of S_{l} on Δ^{l} given by σ · (δ_{k}) = (δ_{σ^{−1}(k)}). The group G acts in the obvious way on the target ∏_{k=1}^{l} X of ev. Let s : Γ ⟶ S_{k} be the group homomorphism satisfying Δγ_{s(γ)(k)}γ = Δγ_{k} ∈ Δ\Γ for all γ ∈ Γ and k ∈ {1, …, l}. Then we obtain an embedding of group

i : Γ ⟶ G γ ↦ (γ_{k}γγ_{s(γ)^{−1}(k)}^{−1})_{k} · s(γ)

such that ev is a Γ-homeomorphism with the given Γ-action on the source and the restricted Γ-action on the target. Hence it suffices to show that ∏_{k=1}^{l} X is G-homotopy equivalent to a dl-dimensional G-CW -complex Y. Recall that a G-CW -structure is the same as a CW -structure such that the G-action permutes the cells and for any g ∈ G and cell e with g · e = e multiplication with g induces the identity on e. Notice that ∏_{k=1}^{l} X has an obvious Δ^{l}-CW -structure. However, the G-action does respect the skeleta and permutes the cells but it happens that multiplication with an element g ∈ G maps a cell to itself but not by the identity so that this does not give the structure of a G-CW -complex on ∏_{k=1}^{l} X. This problem can be solved by using simplicial complexes with equivariant simplicial actions using the fact that the second barycentric subdivision is an equivariant CW -complex but then one has to deal with product of simplicial complexes. We give another approach.

We next construct a nested sequence of G-CW -complexes ∅ = Y_{−1} ⊂ Y_{0} ⊂ … ⊂ Y_{ld} together with G-homotopy equivalence f_{n} : (∏_{k=1}^{l} X)_{n} ⟶ Y_{n} such that Y_{n} is n-dimensional and f_{n} is an extension of f_{n−1} for all n ≥ 0. Then f_{ld} is the desired Γ-homotopy equivalence. The begin n = −1 of the induction is trivial, the induction step from n − 1 to n done as follows.

For p ≥ 0 choose a Δ-push out

∐_{α∈I_{p}} Δ/L_{α} × S^{p−1} ⟶ X_{p−1}
↓  ↓
∐_{α∈I_{p}} Δ/L_{α} × D^{p} ⟶ X_{p}

Define an index set

J_{n} := \{(\underline{i}, \underline{α}) | \underline{i} = (i_{1}, i_{2}, … i_{l}), i_{k} = 0, … n, ∑_{k=1}^{l} i_{k} = n, \underline{α} = (α_{1}, … α_{l}) ∈ I_{i_{1}} × … × I_{i_{l}}\}.

Consider the space

A = ∐_{(\underline{i},\underline{α})∈J_{n}} (∏_{k=1}^{l} Δ/L_{α_{k}} × D^{i_{k}}) .

There is an obvious S_{l}-action on J_{n} given by permutating the coordinates which induces an S_{l}-action on A. Together with the obvious Δ^{l}-action this yields a G-action on A. Next we show that (A, ∂A) carries the structure of a pair of G-CW -complexes with dimension n where ∂A is the topological boundary of A. Fix (\underline{i}, \underline{α}) ∈ J_{n}. Let S′_{l} ⊂ S_{l} be the isotropy group of (\underline{i}, \underline{α}) under the S_{l}-action on J_{n}. Then S′_{l} acts on ∏_{k=1}^{l} D^{i_{k}} by permuting the coordinates. Equip the pair (∏_{k=1}^{l} D^{i_{k}}, ∂ ∏_{k=1}^{l} D^{i_{k}}) with a S′_{l}-CW -complex structure. This induces a G-CW -structure on the pair

∐_{(\underline{i′},\underline{α′})∈J_{n},S_{l}(\underline{i′},\underline{α′})=S_{l}(\underline{i},\underline{α})∈S_{l}\J_{n}} ∏_{k=1}^{l} Δ/L_{α_{k}}×D^{i_{k}}, ∂ ( ∐_{(\underline{i′},\underline{α′})∈J_{n},S_{l}(\underline{i′},\underline{α′})=S_{l}(\underline{i},\underline{α})∈S_{l}\J_{n}} ∏_{k=1}^{l} Δ/L_{α_{k}} × D^{i_{k}} ) .

Since (A, ∂A) is the disjoint union of such pairs, it carries a G-CW -structure. We have the G-push out

∂A ⟶^{q} (∏_{k=1}^{l} X)_{n−1}
↓  ↓
A ⟶ (∏_{k=1}^{l} X)_{n}

By the equivariant cellular approximation theorem [10, II.2] we can choose a G-homotopy h from f_{n−1} ∘ q to a cellular G-map c : ∂A ⟶ Y_{n−1}. Define Y_{n} by the G-push out

∂A ⟶^{c} Y_{n−1}
↓  ↓
A ⟶ Y_{n}

Since c is cellular, (A, ∂A) is a pair of n-dimensional G-CW -complexes and Y_{n−1} an (n − 1)- dimensional G-CW -complex, Y_{n} is an n-dimensional G-CW -complex. Using [15, Lemma 2.13 on page 38] and the G-homotopy h one constructs the desired G-homotopy equivalence f_{n} : (∏_{k=1}^{l} X)_{n} ⟶ Y_{n}. This finishes the proof of Theorem 2.3.1.

2.) Let Z be any (non-equivariant) space and let H ⊂ Γ be a subgroup. We have the isomorphism of Δ-sets

∐_{k=1}^{l} Δ/(γ_{k}Hγ_{k}^{−1} ∩ Δ) ⟶^{≅} i^{∗}Γ/H δ(γ_{k}Hγ_{k}^{−1} ∩ Δ) ↦ δγ_{k}H.

We get from 2.2 the following composition of homeomorphisms

map (Z, (i_{♯}E(Δ, F))^{H}) ≅ map_{Γ} (Γ/H × Z, i_{♯}E(Δ, F)) ≅ map_{Δ} (i^{∗}Γ/H × Z, E(Δ, F)) ≅
map_{Δ} (∐_{k=1}^{l} Δ/(γ_{k}Hγ_{k}^{−1} ∩ Δ) × Z, E(Δ, F)) ≅ ∏_{k=1}^{l} map_{Δ} (Δ/(γ_{k}Hγ_{k}^{−1} ∩ Δ) × Z, E(Δ, F))

≅ ∏_{k=1}^{l} map (Z, E(Δ, F)^{γ_{k}Hγ_{k}^{−1}∩Δ}) .

Notice that H belongs to i_{♯}F if and only if γ_{k}Hγ_{k}^{−1} ∩ Δ belongs to F for k = 1, … l. Hence for any space Z the space map (Z, (i_{♯}E(Δ, F))^{H}) is empty if H does not belong to i_{♯}F and is path-connected otherwise. This finishes the proof of Theorem 2.3.

Theorem 2.3 implies Theorem 2.4. Let i : Δ ⊂ Γ be an inclusion of a subgroup of finite index [Γ : Δ] and F be a family of subgroups of Δ. Suppose that E(Δ, F) has a Δ-CW -model of dimension d. Then E(Γ, i_{♯}F) has a Γ-CW -model of dimension d · [Γ : Δ].

If we take in Theorem 2.4 the family F as the family of finite subgroups of Δ or virtually cyclic subgroup of Δ respectively, then i_{♯}F is the family of finite subgroups of Γ or virtually cyclic subgroup of Γ respectively.

3. Short exact sequences

Let 1 → Δ → Γ → π → 1 be an exact sequence of groups. In this section we want to investigate whether finiteness conditions about the type of a classifying space for FIN for Δ and π carry over to the one of Γ.

Theorem 3.1. Let 1 → Δ ⟶^{i} Γ ⟶^{p} π → 1 be an exact sequence of groups. Suppose that there exists a positive integer d which is an upper bound on the orders of finite subgroups of π. Suppose that \underline{E}Δ has a k-dimensional Δ-CW -model and \underline{E}π has a m-dimensional π-CW -model. Then \underline{E}Γ has a (dk + m)-dimensional Γ-CW -model.

Proof. Let \underline{E}π be a m-dimensional π-CW -model for the classifying space of π for the family of finite subgroups. Let p^{∗}\underline{E}π be the Γ-space obtained from \underline{E}π by restriction with p. We will construct for each −1 ≤ n ≤ m a (dk + n)-dimensional Γ-CW -complex X_{n} and a Γ-map f_{n} : X_{n} ⟶ p^{∗}\underline{E}π_{n} to the n-skeleton of p^{∗}\underline{E}π such that X_{n−1} is a Γ-CW -subcomplex of X_{n}, f_{n} restricted to X_{n−1} is f_{n−1}, for each finite subgroup H ⊂ Γ the map f_{n}^{H} : X_{n}^{H} ⟶ p^{∗}\underline{E}π_{n}^{H} is a homotopy equivalence and all isotropy groups of X_{n} are finite. Then X_{m} will be the desired (dk + m)-dimensional Γ-CW -model for \underline{E}Γ. The begin n = −1 is given by the identity on the empty set, the induction step from n − 1 to n ≥ 0 is done as follows.

Choose a Γ-pushout

∐_{i∈I} Γ/L_{i} × S^{n−1} \underrightarrow{∐_{i∈I} q_{i}} p^{∗}\underline{E}π_{n−1}
↓  ↓
∐_{i∈I} Γ/L_{i} × D^{n} ⟶ p^{∗}\underline{E}π_{n}

There is a canonical projection pr_{i} : Γ ×_{L_{i}} \underline{E}L_{i} ⟶ Γ/L_{i}. Fix a map of sets s : Γ/L_{i} ⟶ Γ whose composition with the projection Γ ⟶ Γ/L_{i} is the identity on Γ/L_{i}. Given an L_{i}-space Y, there is a natural homeomorphism

∐_{w∈Γ/L_{i},s(w)^{−1}Hs(w)⊂L_{i}} Y^{s(w)^{−1}Hs(w)} ⟶^{≅} (Γ ×_{L_{i}} Y)^{H}

which sends y ∈ Y^{s(w)^{−1}Hs(w)} to (s(w), y). Hence pr_{i}^{H} is a homotopy equivalence for all finite subgroups H ⊂ Γ and the isotropy groups of Γ ×_{L_{i}} \underline{E}L_{i} are finite. Since f_{n−1}^{H} is a homotopy equivalence for all finite subgroups H ⊂ Γ, there is a cellular Γ-map

r_{i} : (Γ ×_{L_{i}} \underline{E}L_{i}) × S^{n−1} ⟶ X_{n−1}

together with a Γ-homotopy

h : (Γ ×_{L_{i}} \underline{E}L_{i}) × S^{n−1} × [0, 1] ⟶ p^{∗}\underline{E}π_{n−1}

from f_{n−1} ∘ r_{i} to q_{i} ∘ (pr_{i} × id) [15, Proposition 2.3 on page 35]. Define the following three Γ-spaces by the Γ-push outs

∐_{i∈I}(Γ ×_{L_{i}} \underline{E}L_{i}) × S^{n−1} \underrightarrow{∐_{i∈I} q_{i}∘(pr_{i} × id)} p^{∗}\underline{E}π_{n−1}
↓  ↓
∐_{i∈I}(Γ ×_{L_{i}} \underline{E}L_{i}) × D^{n}  ⟶  X′_{n}

and

∐_{i∈I}(Γ ×_{L_{i}} \underline{E}L_{i}) × S^{n−1} \underrightarrow{∐_{i∈I} f_{n−1}∘r_{i}} p^{∗}\underline{E}π_{n−1}
↓  ↓
∐_{i∈I}(Γ ×_{L_{i}} \underline{E}L_{i}) × D^{n}  ⟶  X′′_{n}

and

∐_{i∈I}(Γ ×_{L_{i}} \underline{E}L_{i}) × S^{n−1} \underrightarrow{∐_{i∈I} r_{i}} X_{n−1}
↓  ↓
∐_{i∈I}(Γ ×_{L_{i}} \underline{E}L_{i}) × D^{n} ⟶ X_{n}

We obtain a Γ-map f′_{n} : X′_{n} ⟶ p^{∗}\underline{E}π_{n} by the Γ-pushout property and the Γ-maps ∐_{i∈I} pr_{i} × id_{D^{n}}, ∐_{i∈I} pr_{i} × id_{S^{n−1}} and id_{p^{∗}\underline{E}π_{n−1}}. Let f′′_{n} : X′′_{n} : X′′_{n} ⟶ X′_{n} be the Γhomotopy equivalence relative p^{∗}\underline{E}π_{n−1} induced by the Γ-homotopy ∐_{i∈I} h_{i}. Let f′′′_{n} : X_{n} ⟶ X′′_{n} be the Γ-map induced by the Γ-maps ∐_{i∈I} id_{Γ×_{L_{i}} \underline{E}L_{i})×D^{n}}, ∐_{i∈I} id_{Γ×_{L_{i}} \underline{E}L_{i})×S^{n−1}} and f_{n−1} and the Γ-pushout property. We define the Γ-map f_{n} : X_{n} ⟶ p^{∗}\underline{E}π_{n} by the composition f′_{n} ∘ f′′_{n} ∘ f′′′_{n}. Since (f′_{n})^{H} and (f′′′_{n})^{H} are homotopy equivalences for finite H ⊂ Γ as they are push outs of homotopy equivalences [15, Lemma 2.13 on page 38] and f′′_{n} is a Γ-homotopy equivalence, f_{n}^{H} is a homotopy equivalence for all finite H ⊂ Γ.

We conclude that the resulting Γ-CW -complex X_{n} has only finite isotropy groups and f^{H} is a homotopy equivalence for each finite H ⊂ Γ. Notice that each cell Γ/L_{i} × D^{n} of p^{∗}\underline{E}π satisfies L_{i} = p^{−1}(H_{i}) for some finite subgroup H_{i} ⊂ π and hence L_{i} contains Δ as subgroup of finite index |H_{i}|. Theorem 2.4 implies that that \underline{E}L_{i} can be choosen as a m|H_{i}|-dimensional L_{i}-CW -complex. Hence the dimension of X_{n} is at most dk + n.

Theorem 3.2. Let 1 → Δ ⟶^{i} Γ ⟶^{p} π → 1 be an exact sequence of groups. Suppose for any finite subgroup π′ ⊂ π and any extension 1 → Δ → Δ′ → π′ → 1 that \underline{E}Δ′ has a finite Δ′-CW -model or a Δ′-CW -model of finite type respectively and suppose that \underline{E}π has a finite π-CW -model or a π-CW -model of finite type respectively. Then \underline{E}Γ has a finite Γ-CW -model or a Γ-CW -model of finite type respectively.

Proof. The proof is exactly the same as the one of Theorem 3.1 except for the very last step. Namely, we have to know that \underline{E}L_{i} has a finite L_{i}-CW -model or a L_{i}-CW -model of finite type respectively and this follows from the assumptions.

Notice that the assumption about \underline{E}Δ′ in Theorem 3.2 would follow from the assumption that \underline{E}Δ has a finite Δ-CW -model or a Δ-CW -model of finite type respectively if Problem 7.2 has an affirmative answer. If π is torsionfree, this assumption about \underline{E}Δ′ reduces to the assumption that there is a finite Δ-CW -model or a Δ-CW -model of finite type for \underline{E}Δ. If Δ is word-hyperbolic, then its Rips complex yields a finite Δ-CW -model for \underline{E}Δ. Moreover a group is word-hyperbolic if it contains a word-hyperbolic subgroup of finite index, because the property word-hyperbolic is a quasi-isometry invariant. It is not hard to check that any virtually poly-cyclic group Δ has a finite Γ-CW -model for \underline{E}Γ. Hence we conclude from Theorem 3.2

Theorem 3.3. Let 1 → Δ ⟶^{i}Γ ⟶^{p}π → 1 be an exact sequence of groups. Suppose that Δ is word-hyperbolic or virtually poly-cyclic. Suppose that \underline{E}π has a finite π-CW -model or a π-CW -model of finite type respectively. Then \underline{E}Γ has a finite Γ-CW -model or a Γ-CW -model of finite type respectively.

4. Classifying spaces of finite type

In this section we deal with the question whether there are Γ-CW -models of finite type for \underline{E}Γ.

Lemma 4.1. If there is a Γ-CW -complex Y of finite type which has only finite isotropy groups and is (non-equivariantly) contractible, then EΓ has a Γ-CW -model of finite type.

Proof. The idea is to replace a cell Γ/H_{i} × D^{n} in Y by the Γ-space Γ ×_{H_{i}} EH_{i} × D^{n}. The construction is an obvious modification of the construction in the proof of Theorem 3.1 using the fact that EH_{i} can be choosen as a H_{i}-CW -complex of finite type. Namely, we construct for n ≥ −1 a Γ-CW -complex X_{n} and a Γ-map f_{n} : X_{n} ⟶ Y_{n} to the n-skeleton of Y such that X_{n} is obtained from X_{n−1} by a Γ-push out of the shape

∐_{i∈I}(Γ ×_{H_{i}} EH_{i}) × S^{n−1} ⟶ X_{n−1}
↓  ↓
∐_{i∈I}(Γ ×_{H_{i}} EH_{i}) × D^{n} ⟶ X_{n}

and f_{n} extends f_{n−1} and is a non-equivariant homotopy equivalence. Then the colimit X = colim_{n→∞} X_{n} is the desired Γ-CW -model for EΓ.

Theorem 4.2. The following statements are equivalent for the group Γ.

1. There is a Γ-CW -model for \underline{E}Γ of finite type;

2. There are only finitely many conjugacy classes of finite subgroups of Γ and for any finite subgroup H ⊂ Γ there is a CW -model for BWH of finite type;

3. There are only finitely many conjugacy classes of finite subgroups of Γ and for any finite subgroup H ⊂ Γ the Weyl group WH is finitely presented and is of type F P_{∞}, i.e. there is a projective ℤWH-resolution of finite type of the trivial ℤWH-module ℤ.

Proof. 1.) ⟹ 2.) Let \underline{E}Γ be a Γ-CW -model of finite type. Let Γ/H_{1}, …, Γ/H_{n} be the finitely many equivariant 0-cells . Then any other equivariant Γ-cell Γ/K × D^{n} in \underline{E}Γ must have the property that K is subconjugated to one of the H_{i}’s because the existence of a Γ-map from Γ/K to Γ/L is equivalent to K being subconjugated to L. Since each H_{i} is finite and has only finitely many distinct subgroups, we conclude that there are only finitely many conjugacy classes of finite subgroups in Γ. Now apply Lemma 1.3 and Lemma 4.1.

2.) ⟹ 1.) Let (H_{1}), …, (H_{r}) be the conjugacy classes of finite subgroups with a numeration such that (H_{i}) is subconjugated to (H_{j}) only if i ≥ j. We construct inductively Γ-CW - complexes X_{0}, X_{1}, …, X_{r} such that X_{0} is empty, X_{n} is obtained from X_{n−1} by attaching cells Γ/H_{n} × D^{m} for m ≥ 0, X_{n} is of finite type and X_{n}^{H_{n}} is contractible. Notice that then X_{r} is a Γ-CW -model of finite type for \underline{E}Γ because X_{n}^{H_{i}} = X_{n−1}^{H_{i}} for i < n.

In the induction step from n − 1 to n it suffices to construct an extension of WH_{n}- CW -complexes X_{n−1}^{H_{n}} ⟶^{i}Z such that Z is contractible, obtained from X_{n−1}^{H_{n}} by attaching free WH_{n}-cells and of finite type. Then one defines X_{n} as the Γ-pushout

Γ ×_{NH_{n}} X_{n−1}^{H_{n}} ⟶^{id_{Γ} ×_{NHn}i} Γ ×_{NH_{n}} Z
j↓  ↓
X_{n−1}  ⟶  X_{n}

where j maps (γ, x) to γx. Notice that X_{n}^{H_{n}} = Z^{H_{n}}. Since each isotropy group of the WH_{n}- space X_{n−1}^{H_{n}} is finite, we can construct a free WH_{n}-CW -complex Y together with a WH_{n}-map h : Y ⟶ X_{n−1}^{H_{n}} which is a (non-equivariant) homotopy equivalence by substituting each cell WH_{n}/L × D^{m} by WH_{n} ×_{L} EL × D^{m}. This construction is an easy modification of the construction in the proof of Theorem 3.1. Since L is finite and hence EL can be choosen as a free L-CW -complex of finite type, Y is of finite type. Let f : Y ⟶ EWH_{n} be the classifying WH_{n}-map. We can choose EWH_{n} of finite type by assumption. Let i : Y ⟶ cyl(f) be the inclusion into the mapping cylinder of f. Notice that cyl(f) is a WH_{n}-CW -model for EWH_{n} of finite type. Define Z by the WH_{n}-pushout

Y ⟶^{i} cyl(f)
h↓  ↓^{g}
X_{n−1}^{H_{n}} ⟶ Z

where i is the inclusion. As h is a (non-equivariant) homotopy equivalence, g is a (nonequivariant) homotopy equivalence. Hence Z is contractible and of finite type.

2.) ⟺ 3.) is a variation of the proof of [6, Theorem VIII.7.1 on page 205] or follows from [15, Proposition 14.9 on page 182].

5. Finitely dominated and finite classifying spaces

In this section we deal with the question whether there are finitely dominated or finite Γ-CW -models for \underline{E}Γ. Recall that a Γ-CW -complex X is called finitely dominated if there are a finite Γ-CW -complex Y and Γ-maps r : Y ⟶ X and i : X ⟶ Y such that r ∘ i is Γ-homotopic to the identity.

Theorem 5.1. The following statements are equivalent for the group Γ.

1. There is a finitely dominated Γ-CW -model for \underline{E}Γ;

2. There are only finitely many conjugacy classes of finite subgroups of Γ and for any finite subgroup H ⊂ Γ the Weyl group WH is finitely presented, is of type F P_{∞} and satisfies condition b(d) for some d ≥ 0.

Proof. A finitely dominated Γ-CW -complex is Γ-homotopy equivalent to a Γ-CW -complex of finite orbit type [15, Proposition 2.12 on page 38]. A Γ-CW -complex X of finite orbit type is finitely dominated if and only if it is Γ-homotopy equivalent to both a Γ-CW -complex Y of finite type and to a finite-dimensional Γ-CW -complex Z. The argument in the proof of [15, Proposition 14.9.a on page 282] applies to the general case. Hence the claim follows from Lemma 1.5, Theorem 1.10 and Theorem 4.2.

If one knows that \underline{E}Γ is finitely dominated, then there is the equivariant finiteness obstruction

õ^{Γ}(\underline{E}Γ) ∈ K̃_{0}(ℤ Or(Γ, FIN)) ≅ ⊕_{\{(H)|H∈FIN \}} K̃_{0}(ℤWH)

whose vanishing is a necessary and sufficient condition for \underline{E}Γ being Γ-homotopy equivalent to a finite Γ-CW -complex ([15, Theorem 14.6 on page 278], [15, Theorem 10.34 on page 196]).

Remark 5.2. If Δ is a word-hyperbolic group, its Rips complex yields a finite Γ-CW - model for \underline{E}Γ. If Γ is a discrete cocompact subgroup of a Lie group G with finitely many components, then G/K with the left Γ-action for K ⊂ G a maximal compact subgroup is a finite Γ-CW -model for \underline{E}Γ [1]. If Γ contains ℤ^{n} as subgroup of finite index, there exists an epimorphism of Γ to a crystallographic group with finite kernel (as pointed out to us by Frank Connolly) and hence an n-dimensional Γ-CW -model for \underline{E}Γ with ℝ^{n} as underlying space. Any virtually poly-cyclic group Γ has a finite Γ-CW -model for \underline{E}Γ. If Γ is Γ_{1} ∗_{Γ_{0}} Γ_{2} for finite groups Γ_{i}, then there is a finite 1-dimensional Γ-CW -model for \underline{E}Γ [19, Theorem 7 in I.4.1 on page 32].

6. Groups with finite virtual cohomological dimension

In this section we investigate the condition b(d) and B(d) of Notation 1.4 and explain how our results specialize in the case of a group of finite virtual cohomological dimension.

Lemma 6.1. If Γ satisfies b(d) or B(d) respectively, then any subgroup Δ of Γ satisfies b(d) or B(d) respectively.

Proof. We begin with b(d). Let M be a ℤΔ-module which is projective over ℤH for all finite subgroups H ⊂ Δ. For a subgroup K ⊂ Γ the double coset formula gives an isomorphism of ℤK-modules

res_{Γ}^{K} ind_{Δ}^{Γ} M ≅ ⊕_{KγΔ∈K\Γ/Δ} ind_{γΔγ^{−1}∩K}^{K} res_{c(γ)} M,  (6.2)

where c(γ) : γΔγ^{−1} ∩ K ⟶ Δ maps δ to γ^{−1}δγ. If K is finite, each subgroup Δ ∩ γ^{−1}Kγ of Δ is finite and hence the ℤγΔγ^{−1} ∩ K-module res_{c(γ)} M is projective. We conclude from 6.2 that the ℤK-module res_{Γ}^{K} ind_{Δ}^{Γ} M is projective for all finite subgroups K ⊂ Γ. Since Γ satisfies b(d) by assumption the ℤΓ-module ind_{Δ}^{Γ} M has a projective resolution of dimension d. If one applies 6.2 to K = Δ, then one concludes that the ℤΔ-module M is a direct summand in res_{Γ}^{Δ} ind_{Δ}^{Γ} M. Hence M has a projective ℤΔ-resolution of dimension d. This finishes the proof for b(d). The claim for B(d) follows since the Weyl group W_{Δ}H of H in Δ is a subgroup of the Weyl group W_{Γ}H of H in Γ for any subgroup H ⊂ Δ.

Recall that a group Γ has virtual cohomological dimension ≤ d if and only if it contains a torsionfree subgroup Δ of finite index such that the trivial ℤΔ-module ℤ has a projective ℤΔ-resolution of dimension d.

Theorem 6.3. Let Γ be a group which contains a torsionfree subgroup Δ of finite index. Then the following assertions are equivalent:

1. Γ satisfies B(d);

2. Γ satisfies b(d);

3. Γ has virtual cohomological dimension ≤ d.

Proof. 1.) ⟹ 2.) is obvious.

2.) ⟹ 3.) The subgroup Δ satisfies b(d) by Lemma 6.1. Since Δ is torsionfree this shows that the virtual cohomological dimension of Γ is less or equal to d.

3.) ⟹ 1.) Suppose that the virtual cohomological dimension of Γ is ≤ d. Next we show that then Γ satisfies b(d). Let M be a ℤΓ-module such that res_{Γ}^{H} M is projective for all finite subgroups H ⊂ Γ. Then M has a d-dimensional projective ℤΓ-resolution by [6, Theorem VI.8.12 on page 152 and Proposition X.5.2 and Theorem X.5.3 on page 287]. Hence Γ satisfies b(d). Next we show that Γ satisfies B(d). Let H ⊂ Γ be a finite subgroup. Then Δ ∩ N_{Γ}H is a subgroup of Γ and hence has virtual cohomological dimension d. Since Δ ∩ H is trivial, Δ ∩ N_{Γ}H is a subgroup in W_{Γ}H of finite index. Hence W_{Γ}H satisfies b(d) by the assertion for b(d) we have just proven above. Therefore Γ satisfies B(d). This finishes the proof of Theorem 6.3.

We rediscover from Theorem 5.1 and Theorem 6.3 the result of [7, Theorem I on page 18] that a group Γ with finite virtual cohomological dimension has a finitely dominated Γ-CW -model for \underline{E}Γ if and only if Γ has only finitely many conjugacy classes of finite subgroups and for each finite subgroup H ⊂ Γ its Weyl group WH is finitely presented and of type F P_{∞}.

Next we improve Theorem 1.10 in the case of groups with finite virtual cohomological dimension. Notice that for such a group there is an upper bound on the length l(H) of finite subgroups H ⊂ Γ.

Theorem 6.4. Let Γ be a group with virtual cohomological dimension ≤ d. Let l ≥ 0 be an integer such that the length l(H) of any finite subgroup H ⊂ Γ is bounded by l. Then there is a Γ-CW -model for \underline{E}Γ such that for any finite subgroup H ⊂ Γ

dim(\underline{E}Γ^{H}) = max\{3, d\} + l − l(H)

holds. In particular \underline{E}Γ has dimension max\{3, d\} + l.

Proof. We want to use the implication (3) ⟹ (1) of Theorem 1.6 where the subgroup Δ(H) ⊂ WH is given by the image of Δ ∩ NH under the projection NH ⟶ WH. Because of Theorem 6.3 the group Γ satisfies B(d). Hence it remains to show

H_{ℤΔ(H)}^{max{3,d}+l−l(H)+1}(EΔ(H) × (C|P(WH)|, |P(WH)|); M) = 0

for any ℤΔ(H)-module M. Notice that Δ(H) is isomorphic to Δ ∩ NH since Δ is torsionfree and H is finite and hence there is a d-dimensional model for EΔ(H). Therefore it suffices to show that |P(WH)| can be choosen to be (l − l(H) − 1)-dimensional because then H_{ℤΔ(H)}^{max{3,d}+l−l(H)}(EΔ(H) × |P(WH)|; M) = 0. This follows from

max{l(K) | K ⊂ WH, K finite } ≤ l − l(H);
max{l(K) | K ⊂ WH, K finite } = dim(|P(WH)|) + 1.

7. Problems

In this section we discuss some open problems. We emphasize that we state most of the problems not because we have a strategy to prove them, but because we lack counterexamples.

Problem 7.1. Which of the results in this paper carry over to a Lie group or more general to a (locally compact) topological group G and the classifying space E(G, K) for the family K of compact subgroups?

Problem 7.2. Let Δ ⊂ Γ be a subgroup of finite index. Suppose that \underline{E}Δ has a Δ-CW - model of finite type or a finite Δ-CW -model respectively. Does then \underline{E}Γ have a Γ-CW -model of finite type or a finite Γ-CW -model respectively?

Remark 7.3. If Problem 7.2 has a positive answer, then the following is true for an exact sequence of groups 1 → Δ → Γ → π → 1. If \underline{E}Δ has a finite Δ-CW -model or a Δ-CW - model of finite type respectively and \underline{E}π has a finite π-CW -model or a π-CW -model of finite type respectively, then \underline{E}Γ has a finite Γ-CW -model or a Γ-CW -model of finite type respectively (cf. Theorem 3.2).

There are some cases where the answer to Problem 7.2 is positive, for example if Δ is word-hyperbolic or virtually poly-cyclic (see Theorem 3.3). Suppose for instance that Γ is torsionfree. There is a normal subgroup Δ′ ⊂ Γ of finite index such that Δ′ ⊂ Δ holds. If BΔ is of finite type or is finite respectively, then BΔ′ is of finite type or is finite respectively. If π is the finite group Γ/Δ′, then we have a fibration BΔ ⟶ BΓ ⟶ Bπ. Since BΔ′ and Bπ are of finite type, BΓ is of finite type [16, Lemma 7.2]. We have shown that BΓ is of finite type if Γ is torsionfree and BΔ is of finite type. If BΔ is finite and Γ is torsionfree, then BΓ is finitely dominated since it is of finite type and by Theorem 2.4 finite-dimensional [15, Proposition 14.9 on page 282]. In order to check that BΓ is finite, one has to compute its finiteness obstruction õ(BΓ) which takes values in K̃_{0}(ℤΓ). Notice that there is the conjecture that K̃_{0}(ℤΓ) vanishes for torsionfree groups Γ. These considerations show that for torsionfree Γ the answer to Problem 7.2 is positive if one asks for “finite type” and very likely to be positive if one asks for “finite”.

There are some reasons to believe that the answer to Problem 7.2 is not always positive or that its proof is difficult. Consider the special case where ϕ : Δ ⟶ Δ is an automorphism of a group Δ with \underline{E}Δ of finite type such that ϕ^{n} = id, the group Γ is the semi-direct product of Δ and ℤ/n with respect to ϕ, and H ⊂ Γ is just ℤ/n. Let CH be the centralizer of H in Γ. Since H is finite and hence has only finitely many automorphisms, CH has finite index in NH. Then CH ∩ Δ is the fixed point set Fix(ϕ) of ϕ and has finite index in NH. Hence because of Theorem 4.2 and [16, Lemma 7.2] a positive answer to Problem 7.2 would imply for any periodic automorphism of a group Δ with \underline{E}Δ of finite type that B Fix(ϕ) has finite type and in particular that Fix(ϕ) is finitely presented.

A positive answer to Problem 7.2 would imply also a positive answer to the following problem by Theorem 4.2.

Problem 7.4. If the group Γ contains a subgroup of finite index Δ which has a Δ-CW - model of finite type for \underline{E}Δ, does then Γ contain only finitely many conjugacy classes of finite subgroups?

To our knowledge Problem 7.4 is open even for torsionfree Δ (see also [6, Lemma IX.13.2 on page 267]).

In view of Theorem 4.2 a positive answer to the next problem implies a positive answer to Problem 7.2 in the case of finite type provided that Γ has only finitely many conjugacy classes of finite subgroups because BΓ is of finite type if for some subgroup Δ of finite index BΔ is of finite type [16, Lemma 7.2].

Problem 7.5. Let Γ be a group such that BΓ is of finite type. Is then BWH of finite type for any finite subgroup H ⊂ Γ?

An algebraic analogoue of Problem 7.5 would be the question whether for a group Γ of type F P_{∞} and any finite subgroup H ⊂ Γ the Weyl group WH is of type F P_{∞}.

Remark 7.6. If Problem 7.5 has a positive answer, then Theorem 4.2 would imply that there is a Γ-CW -model of finite type for \underline{E}Γ if and only if Γ is finitely presented and of type F P_{∞} and there are only finitely many conjugacy classes of finite subgroups in Γ.

Problem 7.7. Let 1 → Δ ⟶^{i}Γ ⟶^{p}π → 1 be an exact sequence of groups. Suppose that there is a Δ-CW -model of finite type for \underline{E}Δ and a Γ-CW -model of finite type for \underline{E}Γ. Is then there a π-CW -model of finite type for \underline{E}π?

The answer to Problem 7.7 is yes if Δ, Γ and π are torsionfree [16, Lemma 7.2]. Since for instance the kernel of the obvious epimorphism from the free group of rank 2 to its abelianization is not even finitely generated, it does not make sense to ask a similiar question where one has information about the classifying spaces for Γ and π and wants to conclude something for the one of Δ.

Problem 7.8. Suppose that there is a finite-dimensional Γ-CW -complex or finite Γ-CW - complex or Γ-CW -complex of finite type respectively whose isotropy groups are all finite and which is contractible. Is then there a finite-dimensional Γ-CW -model or finite Γ-CW -model or Γ-CW -model of finite type respectively for \underline{E}Γ?

The answer to Problem 7.8 is positive in the case finite-dimensional if Γ contains a normal torsionfree subgroup of finite index by Theorem 2.4. A positive answer to Problem 7.8 in the case finite-dimensional is given for groups of type F P_{∞} in [13, Theorem A]. More generally we get a positive answer to Problem 7.8 in the case finite-dimensional from Lemma 1.5 and Theorem 1.10 provided that there is an upper bound l on the length l(H) of the finite subgroups H ⊂ Γ. Problem 7.8 in the case of finite type is equivalent to Problem 7.5 by Lemma 4.1 and Theorem 4.2 provided that there are only finitely many conjugacy classes of finite subgroups of Γ.

Recall that the condition B(d) is necessary for the existence of a d-dimensional Γ-CW - model for \underline{E}Γ (see Lemma 1.5). This is not true for the condition appearing in Theorem 1.6 and Theorem 1.10 that there is an upper bound l on the length l(H) of the finite subgroups H ⊂ Γ (see Example 1.11). Therefore the question arises:

Problem 7.9. For which groups Γ the following is true:

1. For d ≥ 3 the condition B(d) (or the condition b(d) alone) is equivalent to the existence of a d-dimensional Γ-CW -model for \underline{E}Γ;

2. The statements (1), (2) and (3) appearing in Theorem 1.6 are always equivalent (without the assumption that there is an upper bound on the length l(H) of finite subgroups H ⊂ Γ).

Because of Theorem 6.3 Problem 7.9 (1) reduces in the case that Γ contains a torsionfree subgroup of finite index to the problem stated by Brown [5, page 32].

Problem 7.10. For which groups of virtual cohomological dimension ≤ d there exists a d-dimensional Γ-CW -model for \underline{E}Γ.

More generally one may ask

Problem 7.11. Suppose that Γ contains a subgroup Δ of finite index such that there is a ddimensional Δ-CW -model for \underline{E}Δ. Is then there a d-dimensional Γ-CW -model for \underline{E}Γ?

Remark 7.12. A positive answer to Problem 7.11 would imply for an extension of groups 1 → Δ → Γ → π → 1 that \underline{E}Γ has a d + e-dimensional Γ-CW -model if \underline{E}Δ has a ddimensional Δ-CW -model and \underline{E}π has a e-dimensional π-CW -model. In this context the question is interesting whether Γ satisfies B(d+e) if Δ satisfies B(d) and π satisfies B(e).

Problem 7.13. Can one give nice algebraic conditions which ensure the vanishing of the finiteness obstruction õ^{Γ}(\underline{E}Γ) provided that \underline{E}Γ is finitely dominated? Is there at all an example where \underline{E}Γ is finitely dominated but does not admit a finite Γ-CW -model?

Here is a suggestion for a solution of Problem 7.13. A (finite) permutation ℤΓ-module is a ℤΓ-module M which is isomorphic to a (finite) sum of ℤΓ-modules of the shape ℤΓ/H for some finite subgroup H ⊂ Γ.

Problem 7.14. Suppose that Γ satisfies the following condition: There are only finitely many conjugacy classes of finite subgroups in Γ and for each finite subgroup H ⊂ Γ the trivial ℤWH-module ℤ has a finite-dimensional ℤWH-resolution by finite ℤWH-permutation modules. Is then there a finite Γ-CW -model for \underline{E}Γ?

Notice that the condition in Problem 7.14 is necessary for the existence of a finite Γ-CW -model for \underline{E}Γ by Lemma 1.3 and Theorem 5.1 since such a resolution of ℤ is given by the ℤWH-chain complex of \underline{E}WH. Moreover, if the condition is satisfied, \underline{E}Γ is at least finitely dominated by Theorem 5.1. Problem 7.14 has a positive answer if for each finite subgroup H ⊂ Γ and each exact ℤWH-sequence 0 → Q → P_{1} → P_{2} → … → P_{n} → 0 such that Q is finitely generated projective and P_{i} is a finite permutation ℤWH-module for 1 ≤ i ≤ n the ℤWH-module Q is stably free, i.e. becomes free after taking a direct sum with a finitely generated free ℤWH-module. The most optimistic version based on Remark 7.6 and Problem 7.9 would be

Problem 7.15. For which groups are the following statements true for an integer d ≥ 3?

1. There is a d-dimensional Γ-CW -model for \underline{E}Γ if and only Γ satisfies b(d).

2. There is a Γ-CW -model of finite type for \underline{E}Γ if and only if Γ is finitely presented, has only finitely many conjugacy classes of finite subgroups and is of type F P_{∞}.

3. There is a d-dimensional finite Γ-CW -model for \underline{E}Γ if and only if Γ is finitely presented, has only finitely many conjugacy classes of finite subgroups, is of type F P_{∞} and satisfies b(d).

References

[1] H. Abels, Parallizability of proper actions, global K-slices and maximal compact subgroups,, Math. Ann. 212,(1974), 1 - 19

[2] Baum, P., Connes, A., and Higson, N.: “Classifying space for proper actions and K-theory of group C^{∗}-algebras”, in: Doran, R.S. (ed.), C^{∗}-algebras, Contemporary Mathematics 167, 241–291, (1994)

[3] Borel, A. and Serre, J-P.: “Corners and arithmetic groups”, Comment. Math. Helv. 48, 436 - 491 (1974)

[4] Bredon, G.: “Introduction to compact transformation groups”, Academic Press (1972)

[5] Brown, K.S.: “Groups of virtually finite dimension”, in “Homological group theory”, editor: Wall, C.T.C., LMS Lecture Notes Series 36, Cambridge University Press, 27 - 70 (1979)

[6] Brown, K.S.: “Cohomology of groups”, vol. 87 of Graduate texts in Mathematics, Springer (1982)

[7] Connolly, F. and Kozniewski, T.: “Finiteness properties of classifying spaces of proper Γ-actions”, Journal of Pure and Applied Algebra 41, 17–36 (1986)

[8] Davis, J. and Lück, W.: “Spaces over a category and assembly maps in isomorphism conjectures in K-and L-theory”, MPI preprint, to appear in K-Theory (1996)

[9] tom Dieck, T.: “Orbittypen und äquivariante Homologie I”, Arch. Math. 23, 307 - 317 (1972).

[10] tom Dieck, T.: “Transformation groups”, Studies in Math. 8, de Gruyter (1987)

[11] Dunwoody, M.J.: Accessibility and groups of cohomological dimension one; Proc. London Math. Soc 38, 193–215 (1997),

[12] Farrell, F.T. and Jones, L.E.: “Isomorphism conjectures in algebraic K-theory”, J. of the AMS 6, 249 - 298 (1993).

[13] Kropholler, P.H. and Mislin, G.: “Groups acting on finite dimensional spaces with finite stabilizers”, Comment. Math. Helv. 73 (1998), 122-136.

[14] Kropholler, P.H. and Talelli, O: “On a property of fundamental groups of graphs of finite groups”, J. of Pure and Applied Algebra 74, 57 - 59 (1991)

[15] Lück, W.: “Transformation groups and algebraic K-theory”, vol. 1408 of Lecture notes in mathematics, Springer (1989)

[16] Lück, W.: “Hilbert modules and modules over finite von Neumann algebras and applications to L^{2}-invariants”, Math. Annalen 309, 247 - 285 (1997)

[17] Lück, W. and Oliver, R., “The completion theorem in K-theory for proper actions of a discrete group”. preprint (1998)

[18] Serre, J-P.: “Cohomologie des groupes discrets”, Ann. of Math. Studies 70, 77 - 169 (1971)

[19] Serre, J-P.: “Trees”, Springer (1980)

[20] Steenrod, N.E.: “A convenient category of topological spaces”, Mich. Math. J. 14, 133 - 152 (1967)

[21] Wall, C.T.C.: “Finiteness conditions for CW -complexes II ”, Proc. Roy. Soc. London Ser. A 295, 129–139 (1966)

[22] Whitehead, G.W.: “Elements of homotopy theory”, Graduate Texts in Mathematics, Springer (1978)

Address Wolfgang Lück Fachbereich Mathematik und Informatik Westfälische Wilhelms-Universität Münster Einsteinstr. 62 48149 Münster Bundesrepublik Deutschland email: lueck@math.uni-muenster.de FAX: 0251 8338370 internet: http://wwwmath.uni-muenster.de/math/u/lueck/

Version of September 14, 1998