Survey on Classifying Spaces for Families of Subgroups

Wolfgang Lück^{∗} Fachbereich Mathematik Universität Münster Einsteinstr. 62 48149 Münster Germany

June 13, 2004

Abstract

We define for a topological group G and a family of subgroups F two versions for the classifying space for the family F, the G-CW -version E_{F}(G) and the numerable G-space version J_{F}(G). They agree if G is discrete, or if G is a Lie group and each element in F compact, or if F is the family of compact subgroups. We discuss special geometric models for these spaces for the family of compact open groups in special cases such as almost connected groups G and word hyperbolic groups G. We deal with the question whether there are finite models, models of finite type, finite dimensional models. We also discuss the relevance of these spaces for the Baum-Connes Conjecture about the topological K-theory of the reduced group C^{∗}-algebra, for the Farrell-Jones Conjecture about the algebraic Kand L-theory of group rings, for Completion Theorems and for classifying spaces for equivariant vector bundles and for other situations. Key words: Family of subgroups, classifying spaces, Mathematics Subject Classification 2000: 55R35, 57S99, 20F65, 18G99.

0 Introduction

We define for a topological group G and a family of subgroups F two versions for the classifying space for the family F, the G-CW -version E_{F}(G) and the numerable G-space version J_{F}(G). They agree, if G is discrete, or if G is a Lie group and each element in F is compact, or if each element in F is open, or if F is the family of compact subgroups, but not in general.

∗email: lueck@math.uni-muenster.de www: http://www.math.uni-muenster.de/u/lueck/ FAX: 49 251 8338370

One motivation for the study of these classifying spaces comes from the fact that they appear in the Baum-Connes Conjecture about the topological K-theory of the reduced group C^{∗}-algebra and in the Farrell-Jones Conjecture about the algebraic K- and L-theory of group rings and that they play a role in the formulations and constructions concerning Completion Theorems and classifying spaces for equivariant vector bundles and other situations. Because of the Baum-Connes Conjecture and the Farrell-Jones Conjecture the computation of the relevant K- and L-groups can be reduced to the computation of certain equivariant homology groups applied to these classifying spaces for the family of finite subgroups or the family of virtually cyclic subgroups. Therefore it is important to have nice geometric models for these spaces E_{F}(G) and J_{F}(G) and in particular for the orbit space G\E_{FIN} (G).

The space E_{F}(G) has for the family of compact open subgroups or of finite subgroups nice geometric models for instance in the cases, where G is an almost connected group G, where G is a discrete subgroup of a connected Lie group, where G is a word hyperbolic group, arithmetic group, mapping class group, onerelator group and so on. Models are given by symmetric spaces, Teichmüller spaces, outer space, Rips complexes, buildings, trees and so on. On the other hand one can construct for any CW -complex X a discrete group G such that X and G\E_{FIN} (G) are homotopy equivalent.

We deal with the question whether there are finite models, models of finite type, finite dimensional models. In some sense the algebra of a discrete group G is reflected in the geometry of the spaces E_{FIN} (G). For torsionfree discrete groups E_{FIN} (G) is the same as EG. For discrete groups with torsion the space E_{FIN} (G) seems to carry relevant information which is not present in EG. For instance for a discrete group with torsion EG can never have a finite dimensional model, whereas this is possible for E_{FIN} (G) and the minimal dimension is related to the notion of virtual cohomological dimension.

The space J_{COM}(G) associated to the family of compact subgroups is sometimes also called the classifying space for proper group actions. We will abbreviate it as \underline{J}G. Analogously we often write \underline{E}G instead of E_{COM}(G). Sometimes the abbreviation \underline{E}G is used in the literature, especially in connection with the Baum-Connes Conjecture, for the G-space denoted in this article by \underline{J}G = J_{COM}(G). This does not really matter since we will show that the up to G-homotopy unique G-map \underline{E}G → \underline{J}G is a G-homotopy equivalence.

A reader, who is only interested in discrete groups, can skip Sections 2 and 3 completely.

Group means always locally compact Hausdorff topological group. Examples are discrete groups and Lie groups but we will also consider other groups. Space always means Hausdorff space. Subgroups are always assumed to be closed. Notice that isotropy groups of G-spaces are automatically closed. A map is always understood to be continuous.

The author is grateful to Britta Nucinkis, Ian Leary and Guido Mislin for useful comments.

Contents

0 Introduction 1

1 G-CW -Complex-Version 4 1.1 Basics about G-CW -Complexes . . . . . . . . . . . . . . . . . . . 4 1.2 The G-CW -Version for the Classifying Space for a Family …. 7

2 Numerable G-Space-Version 8

3 Comparison of the Two Versions 10

4 Special Models 18 4.1 Operator Theoretic Model . . . . . . . . . . . . . . . . . . . . . . 18 4.2 Almost Connected Groups . . . . . . . . . . . . . . . . . . . . . . 19 4.3 Actions on Simply Connected Non-Positively Curved Manifolds . 19 4.4 Actions on CAT(0)-spaces . . . . . . . . . . . . . . . . . . . . . . 19 4.5 Actions on Trees and Graphs of Groups . . . . . . . . . . . . . . 20 4.6 Affine Buildings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.7 The Rips Complex of a Word-Hyperbolic Group . . . . . . . . . 24 4.8 Arithmetic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.9 Outer Automorphism Groups of Free groups . . . . . . . . . . . . 25 4.10 Mapping Class groups . . . . . . . . . . . . . . . . . . . . . . . . 26 4.11 Groups with Appropriate Maximal Finite Subgroups . . . . . . . 26 4.12 One-Relator Groups . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.13 Special Linear Groups of (2,2)-Matrices . . . . . . . . . . . . . . 28 4.14 Manifold Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5 Finiteness Conditions 29 5.1 Review of Finiteness Conditions on BG . . . . . . . . . . . . . . 30 5.2 Modules over the Orbit Category . . . . . . . . . . . . . . . . . . 31 5.3 Reduction from Topological Groups to Discrete Groups . . . . . 32 5.4 Poset of Finite Subgroups . . . . . . . . . . . . . . . . . . . . . . 34 5.5 Extensions of Groups . . . . . . . . . . . . . . . . . . . . . . . . . 36 5.6 One-Dimensional Models for \underline{E}G . . . . . . . . . . . . . . . . . . 36 5.7 Groups of Finite Virtual Dimension . . . . . . . . . . . . . . . . 37 5.8 Counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

6 The Orbit Space of \underline{E}G 40

7 Relevance and Applications of Classifying Spaces for Families 42 7.1 Baum-Connes Conjecture . . . . . . . . . . . . . . . . . . . . . . 42 7.2 Farrell-Jones Conjecture . . . . . . . . . . . . . . . . . . . . . . . 43 7.3 Completion Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 44 7.4 Classifying Spaces for Equivariant Bundles . . . . . . . . . . . . . 44 7.5 Equivariant Homology and Cohomology . . . . . . . . . . . . . . 44 8 Computations using Classifying Spaces for Families 44

8.1 Group Homology . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 8.2 Topological K-Theory of Group C^{∗}-Algebras . . . . . . . . . . . 45 8.3 Algebraic K-and L-Theory of Group Rings . . . . . . . . . . . . 46

References 51

Notation 58

Index 59

1 G-CW -Complex-Version

In this section we explain the G-CW -complex version of the classifying space for a family F of subgroups of a group G.

1.1 Basics about G-CW -Complexes

Definition 1.1 (G-CW -complex). A G-CW -complex X is a G-space together with a G-invariant filtration

∅ = X_{−1} ⊆ X_{0} ⊂ X_{1} ⊆ … ⊆ X_{n} ⊆ … ⊆ ⋃_{n≥0} X_{n} = X

such that X carries the colimit topology with respect to this filtration (i.e. a set C ⊆ X is closed if and only if C ∩ X_{n} is closed in X_{n} for all n ≥ 0) and X_{n} is obtained from X_{n−1} for each n ≥ 0 by attaching equivariant n-dimensional cells, i.e. there exists a G-pushout

∐_{i∈I_{n}} G/H_{i} × S^{n−1} \underrightarrow{∐_{i∈In} q_{i}^{n}} X_{n−1}
↓  ↓
∐_{i∈I_{n}} G/H_{i} × D^{n} \overrightarrow{∐_{i∈In} Q_{i}^{n}} X_{n}

The space X_{n} is called the n-skeleton of X. Notice that only the filtration by skeletons belongs to the G-CW -structure but not the G-pushouts, only their existence is required. An equivariant open n-dimensional cell is a G-component of X_{n} − X_{n−1}, i.e. the preimage of a path component of G\(X_{n} − X_{n−1}). The closure of an equivariant open n-dimensional cell is called an equivariant closed n-dimensional cell. If one has chosen the G-pushouts in Definition 1.1, then the equivariant open n-dimensional cells are the G-subspaces Q_{i}(G/H_{i} × (D^{n} − S^{n−1})) and the equivariant closed n-dimensional cells are the G-subspaces Q_{i}(G/H_{i} × D^{n}).

Remark 1.2 (Proper G-CW -complexes). A G-space X is called proper if for each pair of points x and y in X there are open neighborhoods V_{x} of x and W_{y} of y in X such that the closure of the subset {g ∈ G | gV_{x} ∩ W_{y} ≠ ∅} of G is compact. A G-CW -complex X is proper if and only if all its isotropy groups are compact [48, Theorem 1.23]. In particular a free G-CW -complex is always proper. However, not every free G-space is proper.

Remark 1.3 (G-CW -complexes with open isotropy groups). Let X be a G-space with G-invariant filtration

∅ = X_{−1} ⊆ X_{0} ⊆ X_{1} ⊆ … ⊆ X_{n} ⊆ … ⊆ ⋃_{n≥0} X_{n} = X.

Then the following assertions are equivalent. i.) Every isotropy group of X is open and the filtration above yields a G-CW -structure on X. ii.) The filtration above yields a (non-equivariant) CW -structure on X such that each open cell e ⊆ X and each g ∈ G with ge ∩ e ≠ ∅ left multiplication with g induces the identity on e.

In particular we conclude for a discrete group G that a G-CW -complex X is the same as a CW -complex X with G-action such that for each open cell e ⊆ X and each g ∈ G with ge ∩ e ≠ ∅ left multiplication with g induces the identity on e.

Example 1.4 (Lie groups acting properly and smoothly on manifolds). If G is a Lie group and M is a (smooth) proper G-manifold, then an equivariant smooth triangulation induces a G-CW -structure on M. For the proof and for equivariant smooth triangulations we refer to [36, Theorem I and II].

Example 1.5 (Simplicial actions). Let X be a simplicial complex on which the group G acts by simplicial automorphisms. Then all isotropy groups are closed and open. Moreover, G acts also on the barycentric subdivision X′ by simplicial automorphisms. The filtration of the barycentric subdivision X′ by the simplicial n-skeleton yields the structure of a G-CW -complex what is not necessarily true for X.

A G-space is called cocompact if G\X is compact. A G-CW -complex X is finite if X has only finitely many equivariant cells. A G-CW -complex is finite if and only if it is cocompact. A G-CW -complex X is of finite type if each nskeleton is finite. It is called of dimension ≤ n if X = X_{n} and finite dimensional if it is of dimension ≤ n for some integer n. A free G-CW -complex X is the same as a G-principal bundle X → Y over a CW -complex Y (see Remark 2.8).

Theorem 1.6 (Whitehead Theorem for Families). Let f: Y → Z be a G-map of G-spaces. Let F be a set of (closed) subgroups of G which is closed under conjugation. Then the following assertions are equivalent:

(i) For any G-CW -complex X, whose isotropy groups belong to F, the map

induced by f

f_{∗}: [X, Y]^{G} → [X, Z]^{G}, [g] ↦ [g ∘ f]

between the set of G-homotopy classes of G-maps is bijective; (ii) For any H ∈ F the map f^{H}: Y^{H} → Z^{H} is a weak homotopy equivalence

i.e. the map π_{n}(f^{H}, y): π_{n}(Y^{H}, y) → π_{n}(Z^{H}, f^{H}(y)) is bijective for any base point y ∈ Y^{H} and n ∈ ℤ, n ≥ 0.

Proof. (i) ⇒ (ii) Evaluation at 1H induces for any CW -complex A (equipped with the trivial G-action) a bijection [G/H × A, Y]^{G} ⟶^{≅} [A, Y^{H}]. Hence for any CW -complex A the map f^{H} induces a bijection

(f^{H})_{∗}: [A, Y^{H}] → [A, Z^{H}], [g] → [g ∘ f^{H}].

This is equivalent to f^{H} being a weak homotopy equivalence by the classical non-equivariant Whitehead Theorem [84, Theorem 7.17 in Chapter IV.7 on page 182].

(ii) ⇒ (i) We only give the proof in the case, where Z is G/G since this is the main important case for us and the basic idea becomes already clear. The general case is treated for instance in [78, Proposition II.2.6 on page 107]. We have to show for any G-CW -complex X such that two G-maps f_{0}, f_{1} : X → Y are G-homotopic provided that for any isotropy group H of X the H-fixed point set Y^{H} is weakly contractible i.e. π_{n}(Y^{H}, y) consists of one element for all base points y ∈ Y^{H}. Since X is colim_{n→∞} X_{n} it suffices to construct inductively over n G-homotopies h[n]: X_{n} ×[0, 1] → Z such that h[n]_{i} = f_{i} holds for i = 0, 1 and h[n]|_{X_{n−1}×[0,1]} = h[n − 1]. The induction beginning n = −1 is trivial because of X_{−1} = ∅, the induction step from n − 1 to n ≥ 0 done as follows. Fix a G-pushout

∐_{i∈I_{n}} G/H_{i} × S^{n−1} \underrightarrow{∐_{i∈In} q_{i}^{n}} X_{n−1}
↓  ↓
∐_{i∈I_{n}} G/H_{i} × D^{n} \overrightarrow{∐_{i∈In} Q_{i}^{n}} X_{n}

One easily checks that the desired G-homotopy h[n] exists if and only if we can find for each i ∈ I an extension of the G-map

f_{0} ∘ Q_{i}^{n} ∪ f_{1} ∘ Q_{i}^{n} ∪ h[n − 1] ∘ (q_{i}^{n} × id_{[0,1]}):

G/H_{i} × D^{n} × {0} ∪ G/H_{i} × D^{n} × {1} ∪ G/H_{i} × S^{n−1} × [0, 1] → Y

to a G-map G/H_{i} × D^{n} × [0, 1] → Y. This is the same problem as extending the (non-equivariant) map D^{n} × {0} ∪ D^{n} × {1} ∪ S^{n−1} × [0, 1] → Y, which is given by restricting the G-map above to 1H_{i}, to a (non-equivariant) map D^{n} × [0, 1] → Y^{H_{i}}. Such an extension exists since Y^{H_{i}} is weakly contractible. This finishes the proof of Theorem 1.6.

A G-map f: X → Y of G-CW -complexes is a G-homotopy equivalence if and only if for any subgroup H ⊆ G which occurs as isotropy group of X or Y the induced map f^{H}: X^{H} → Y^{H} is a weak homotopy equivalence. This follows from the Whitehead Theorem for Families 1.6 above.

A G-map of G-CW -complexes f: X → Y is cellular if f(X_{n}) ⊆ Y_{n} holds for all n ≥ 0. There is an equivariant version of the Cellular Approximation Theorem, namely, every G-map of G-CW -complexes is G-homotopic to a cellular one and each G-homotopy between cellular G-maps can be replaced by a cellular G-homotopy [78, Theorem II.2.1 on page 104].

1.2 The G-CW -Version for the Classifying Space for a Family

Definition 1.7 (Family of subgroups). A family F of subgroups of G is a set of (closed) subgroups of G which is closed under conjugation and finite intersections.

Examples for F are

TR  = {trivial subgroup};
FIN  = {finite subgroups};
VCYC  = {virtually cyclic subgroups};
COM  = {compact subgroups};
COMOP = {compact open subgroups};
ALL  = {all subgroups}.

Definition 1.8 (Classifying G-CW -complex for a family of subgroups). Let F be a family of subgroups of G. A model E_{F}(G) for the classifying G-CW -complex for the family F of subgroups is a G-CW -complex E_{F}(G) which has the following properties: i.) All isotropy groups of E_{F}(G) belong to F. ii.) For any G-CW -complex Y, whose isotropy groups belong to F, there is up to G-homotopy precisely one G-map Y → X.

We abbreviate \underline{E}G := E_{COM}(G) and call it the universal G-CW -complex for proper G-actions.

In other words, E_{F}(G) is a terminal object in the G-homotopy category of G-CW -complexes, whose isotropy groups belong to F. In particular two models for E_{F}(G) are G-homotopy equivalent and for two families F_{0} ⊆ F_{1} there is up to G-homotopy precisely one G-map E_{F_{0}}(G) → E_{F_{1}}(G).

Theorem 1.9 (Homotopy characterization of E_{F}(G)). Let F be a family of subgroups.

(i) There exists a model for E_{F}(G) for any family F;

(ii) A G-CW -complex X is a model for E_{F}(G) if and only if all its isotropy groups belong to F and for each H ∈ F the H-fixed point set X^{H} is weakly contractible.

Proof. (i) A model can be obtained by attaching equivariant cells G/H ×D^{n} for all H ∈ F to make the H-fixed point sets weakly contractible. See for instance [48, Proposition 2.3 on page 35].

(ii) This follows from the Whitehead Theorem for Families 1.6 applied to f: X → G/G.

A model for E_{ALL}(G) is G/G. In Section 4 we will give many interesting geometric models for classifying spaces E_{F}(G), in particular for the case, where G is discrete and F = FIN or, more generally, where G is a (locally compact topological Hausdorff) group and F = COM. In some sense \underline{E}G = E_{COM}(G) is the most interesting case.

2 Numerable G-Space-Version

In this section we explain the numerable G-space version of the classifying space for a family F of subgroups of group G.

Definition 2.1 (F-numerable G-space). A F-numerable G-space is a Gspace, for which there exists an open covering {U_{i} | i ∈ I} by G-subspaces such that there is for each i ∈ I a G-map U_{i} → G/G_{i} for some G_{i} ∈ F and there is a locally finite partition of unity {e_{i} | i ∈ I} subordinate to {U_{i} | i ∈ I} by G-invariant functions e_{i}: X → [0, 1].

Notice that we do not demand that the isotropy groups of a F-numerable G-space belong to F. If f: X → Y is a G-map and Y is F-numerable, then X is also F-numerable.

Lemma 2.2. Let F be a family. Then a G-CW -complex is F-numerable if each isotropy group is a subgroup of an element in F.

Proof. This follows from the Slice Theorem for G-CW -complexes [48, Theorem 1.37] and the fact that G\X is a CW -complex and hence paracompact [64].

Definition 2.3 (Classifying numerable G-space for a family of subgroups). Let F be a family of subgroups of G. A model J_{F}(G) for the classifying numerable G-space for the family F of subgroups is a G-space which has the following properties: i.) J_{F}(G) is F-numerable. ii.) For any F-numerable G-space X there is up to G-homotopy precisely one G-map X → J_{F}(G).

We abbreviate \underline{J}G := J_{COM}(G) and call it the universal numerable G-space for proper G-actions, or briefly the universal space for proper G-actions.

In other words, J_{F}(G) is a terminal object in the G-homotopy category of F-numerable G-spaces. In particular two models for J_{F}(G) are G-homotopy equivalent, and for two families F_{0} ⊆ F_{1} there is up to G-homotopy precisely one G-map J_{F_{0}}(G) → J_{F_{1}}(G).

Remark 2.4 (Proper G-spaces). A COM-numerable G-space X is proper. Not every proper G-space is COM-numerable. But a G-CW -complex X is proper if and only if it is COM-numerable (see Lemma 2.2).

Theorem 2.5 (Homotopy characterization of J_{F}(G)). Let F be a family of subgroups.

(i) For any family F there exists a model for J_{F}(G) whose isotropy groups belong to F;

(ii) Let X be a F-numerable G-space. Equip X × X with the diagonal action and let pr_{i}: X × X → X be the projection onto the i-th factor for i = 1, 2. Then X is a model for J_{F}(G) if and only if for each H ∈ F there is x ∈ X with H ⊆ G_{x} and pr_{1} and pr_{2} are G-homotopic.

(iii) For H ∈ F the H-fixed point set J_{F}(G)^{H} is contractible.

Proof. (i) A model for J_{F}(G) is constructed in [78, Theorem I.6.6. on page 47] and [7, Appendix 1], namely, as the infinite join ∗_{n=1}^{∞}Z for Z = ∐_{H∈F} G/H. There G is assumed to be compact but the proof goes through for locally compact topological Hausdorff groups. The isotropy groups are finite intersections of the isotropy groups appearing in Z and hence belong to F.

(ii) Let X be a model for the classifying space J_{F}(G) for F. Then X × X with the diagonal G-action is a F-numerable G-space. Hence pr_{1} and pr_{2} are G-homotopic by the universal property. Since for any H ∈ F the G-space G/H is F-numerable, there must exist a G-map G/H → X by the universal property of J_{F}(G). If x is the image under this map of 1H, then H ⊆ G_{x}.

Suppose that X is a G-space such that for each H ∈ F there is x ∈ X with H ⊆ G_{x} and pr_{1} and pr_{2} are G-homotopic. We want to show that then X is a model for J_{F}(G). Let f_{0}, f_{1} : Y → X be two G-maps. Since pr_{i} ∘(f_{0} × f_{1}) = f_{i} holds for i = 0, 1, f_{0} and f_{1} are G-homotopic. It remains to show for any F-numerable G-space Y that there exists a G-map Y → X. Because of the universal property of J_{F}(G) it suffices to do this in the case, where Y = ∗_{n=1}^{∞}L for L = ∐_{H∈F} G/H. By assumption there is a G-map L → X. Analogous to the construction in [7, Appendix 2] one uses a G-homotopy from pr_{1} to pr_{2} to construct a G-map ∗_{n=1}^{∞}L → X.

(iii) Restricting to 1H yields a bijection

[G/H × J_{F}(G)^{H}, J_{F}(G)]^{G} ⟶^{≅} [J_{F}(G)^{H}, J_{F}(G)^{H}],

where we consider X^{H} as a G-space with trivial G action. Since G/H × X^{H} is a F-numerable G-space, [J_{F}(G)^{H}, J_{F}(G)^{H}] consists of one element. Hence J_{F}(G)^{H} is contractible.

Remark 2.6. We do not know whether the converse of Theorem 2.5 (iii) is true, i.e. whether a F-numerable G-space X is a model for J_{F}(G) if X^{H} is contractible for each H ∈ F.

Example 2.7 (Numerable G-principal bundles). A numerable (locally trivial) G-principal bundle p: E → B consists by definition of a TR-numerable Gspace E, a space B with trivial action and a surjective G-map p : E → B such that the induced map G\E → B is a homeomorphism. A numerable G-principal bundle p: EG → BG is universal if and only if each numerable G-bundle admits a G-bundle map to p and two such G-bundle maps are G-bundle homotopic. A numerable G-principal bundle is universal if and only if E is contractible. This follows from [26, 7.5 and 7.7]. More information about numerable G-principal bundles can be found for instance in [35, Section 9 in Chapter 4] [78, Chapter I Section 8].

If p: E → B is a universal numerable G-principal bundle, then E is a model for J_{TR}(G). Conversely, J_{TR}(G) → G\J_{TR}(G) is a model for the universal numerable G-principal bundle. We conclude that a TR-numerable G-space X is a model for J_{TR}(G) if and only if X is contractible (compare Remark 2.6).

Remark 2.8 (G-Principal bundles over CW -complexes). Let p: E → B be a (locally trivial) G-principal bundle over a CW -complex. Since any CW - complex is paracompact [64], it is automatically a numerable G-principal bundle. The CW -complex structure on B pulls back to G-CW -structure on E [48, 1.25 on page 18]. Conversely, if E is a free G-CW-complex, then E → G\E is a numerable G-principal bundle over a CW -complex by Lemma 2.2

The classifying bundle map from p above to J_{TR}(G) → G\J_{TR}(G) lifts to a G-bundle map from p to E_{TR}(G) → G\E_{TR}(G) and two such G-bundle maps from p to E_{TR}(G) → G\E_{TR}(G) are G-bundle homotopic. Hence for G-principal bundles over CW -complexes one can use E_{TR}(G) → G\E_{TR}(G) as the universal object.

We will compare the spaces E_{F}(G) and J_{F}(G) in Section 3. In Section 4 we will give many interesting geometric models for E_{F}(G) and J_{F}(G) in particular in the case F = COM. In some sense \underline{J}G = J_{COM}(G) is the most interesting case.

3 Comparison of the Two Versions

In this section we compare the two classifying spaces E_{F}(G) and J_{F}(G) and show that the two classifying spaces \underline{E}G and \underline{J}G agree up to G-homotopy equivalence.

Since E_{F}(G) is a F-numerable space by Lemma 2.2, there is up to Ghomotopy precisely one G-map

u: E_{F}(G) → J_{F}(G).  (3.1)

Lemma 3.2. The following assertions are equivalent for a family F of subgroups of G:

(i) The map u: E_{F}(G) → J_{F}(G) defined in 3.1 is a G-homotopy equivalence;

(ii) The G-spaces E_{F}(G) and J_{F}(G) are G-homotopy equivalent;

(iii) The G-space J_{F}(G) is G-homotopy equivalent to a G-CW -complex, whose isotropy groups belong to F; (iv) There exists a G-map J_{F}(G) → Y to a G-CW -complex Y, whose isotropy

groups belong to F;

Proof. This follows from the universal properties of E_{F}(G) and J_{F}(G).

Lemma 3.3. Suppose either that every element H ∈ F is an open (and closed) subgroup of G or that G is a Lie group and F ⊆ COM. Then the map u: E_{F}(G) → J_{F}(G) defined in 3.1 is a G-homotopy equivalence.

Proof. We have to inspect the construction in [78, Lemma 6.13 in Chapter I on page 49] and will use the same notation as in that paper. Let Z be a Fnumerable G-space. Let X = ∐_{H∈F} G/H. Then ∗_{n=1}^{∞}X is a model for J_{F}(G) by [78, Lemma 6.6 in Chapter I on page 47]. We inspect the construction of a G-map f: Z → ∗_{n=1}^{∞}X. One constructs a countable covering {U_{n} | n = 1, 2, …} of Z by G-invariant open subsets of Z together with a locally finite subordinate partition of unity {v_{n} | n = 1, 2, …} by G-invariant functions v_{n}: Z → [0, 1] and G-maps ϕ_{n}: U_{n} → X. Then one obtains a G-map

f: Z → ∗_{n=1}^{∞}X, z ↦ (v_{1}(z)ϕ_{1}(z), v_{2}(z)ϕ_{2}(z), …),

where v_{n}(z)ϕ_{n}(z) means 0x for any x ∈ X if z ∉ U_{n}. Let i_{k}: ∗_{n=1}^{k} X → ∗_{n=1}^{∞}X and j_{k}: ∗_{n=1}^{k} X → ∗_{n=1}^{k+1}X be the obvious inclusions. Denote by α_{k}: ∗_{n=1}^{k} X → colim_{k→∞} ∗_{n=1}^{k}X the structure map and by i: colim_{k→∞} ∗_{n=1}^{k}X → ∗_{n=1}^{∞}X the map induced by the system {i_{k} | k = 1, 2, …}. This G-map is a (continuous) bijective G-map but not necessarily a G-homeomorphism. Since the partition {v_{n} | n = 1, 2, …} is locally finite, we can find for each z ∈ Z an open Ginvariant neighborhood W_{z} of z in Z and a positive integer k_{z} such that v_{n} vanishes on W_{z} for n > k_{z}. Define a map

f′_{z}: W_{z} → ∗_{n=1}^{k_{z}}X, z ↦ (v_{1}(z)ϕ_{1}(z), v_{2}(z)ϕ_{2}(z), …, v_{k_{z}}(z)ϕ_{k_{z}}(z)).

Then α_{k_{z}} ∘f′_{z}: W_{z} → colim_{k→∞} ∗_{n=1}^{k}X is a well-defined G-map whose composition with i: colim_{k→∞} ∗_{n=1}^{k}X → ∗_{n=1}^{∞}X is f|_{W_{z}}. Hence the system of the maps α_{k_{z}} ∘ f′_{z} defines a G-map

f′: Z → colim_{k→∞} ∗_{n=1}^{k}X

such that i ∘ f′ = f holds.

Let

Δ_{n−1} = {(t_{1}, t_{2} … t_{n}) | t_{i} ∈ [0, 1], ∑_{i=1}^{n} t_{i} = 1} ⊆ ∏_{n=1}^{k}[0, 1]

be the standard (n − 1)-simplex. Let

p: (∏_{n=1}^{k} X) × Δ_{n} → ∗_{n=1}^{k}X, (x_{1}, …, x_{n}), (t_{1}, …, t_{n}) ↦ (t_{1}x_{1}, …, t_{n}x_{n})

be the obvious projection. It is a surjective continuous map but in general not an identification. Let \overline{∗}_{n=1}^{k}X be the topological space whose underlying set is the same as for ∗_{n=1}^{k}X but whose topology is the quotient topology with respect to p. The identity induces a (continuous) map \overline{∗}_{n=1}^{k}X → ∗_{n=1}^{k}X which is not a homeomorphism in general. Choose for n ≥ 1 a (continuous) function ϕ_{n}: [0, 1] → [0, 1] which satisfies ϕ_{n}^{−1}(0) = [0, 4^{−n}]. Define

u_{k}: ∗_{n=1}^{k} X → \overline{∗}_{n=1}^{k}X,

(t_{n}x_{n} | n = 1, …, k) ↦ ( \frac{ϕ_{n}(t_{n})}{∑_{n=1}^{k} ϕ_{n}(t_{n})}x_{n}| n = 1, …, k) .

It is not hard to check that this G-map is continuous. If \overline{j}_{k}: \overline{∗}_{n=1}^{k}X → \overline{∗}_{n=1}^{k+1}X is the obvious inclusion, we have u_{k+1} ∘ j_{k} = \overline{j}_{k} ∘ u_{k} for all k ≥ 1. Hence the system of the maps u_{k} induces a G-map

u: colim_{k→∞} ∗_{n=1}^{k}X → colim_{k→∞} \overline{∗}_{n=1}^{k}X.

Next we want to show that each G-space \overline{∗}_{n=1}^{k}X has the G-homotopy type of a G-CW -complex, whose isotropy groups belong to F. We first show that \overline{∗}_{n=1}^{k}X is a (∏_{n=1}^{k} G)-CW -complex. It suffices to treat the case k = 2, the general case follows by induction over k. We can rewrite X\overline{∗}X as a G × Gpushout

X × X ⟶^{i_{1}} CX × X
i_{2}↓  ↓
X × CX ⟶ X\overline{∗}X

where CX is the cone over X and i_{1} and i_{2} are the obvious inclusions. The product of two finite dimensional G-CW -complexes is in a canonical way a finite dimensional (G × G)-CW -complex, and, if (B, A) is a G-CW -pair, C a G-CW -complex and f: B → C is a cellular G-map, then A ∪_{f} C inherits a G-CW -structure in a canonical way. Thus X\overline{∗}X inherits a (G × G)-CW -complex structure.

The problem is now to decide whether the (∏_{n=1}^{k} G)-CW -complex \overline{∗}_{n=1}^{k}X regarded as a G-space by the diagonal action has the G-homotopy type of a G-CW -complex. If each H ∈ F is open, then each isotropy group of the G-space ∗_{n=1}^{k}X is open and we conclude from Remark 1.3 that \overline{∗}_{n=1}^{k}X with the diagonal G-action is a G-CW -complex Suppose that G is a Lie group and each H ∈ F is compact. Example 1.4 implies that for any compact subgroup K ⊆ ∏_{n=1}^{k} G the space (∏_{n=1}^{k} G) /K regarded as G-space by the diagonal action has the G-homotopy type of a G-CW -complex. We conclude from [48, Lemma 7.4 on page 121] that \overline{∗}_{n=1}^{k}X with the diagonal G-action has the G-homotopy type of a G-CW -complex. The isotropy groups \overline{∗}_{n=1}^{k}X belong to F since F is closed under finite intersections and conjugation. It is not hard to check that each Gmap \overline{j}_{k} is a G-cofibration. Hence colim_{k→∞} \overline{∗}_{n=1}^{k}X has the G-homotopy type of a G-CW -complex, whose isotropy groups belong to F.

Thus we have shown for every F-numerable G-space Z that it admits a Gmap to a G-CW -complex whose isotropy groups belong to F. Now Lemma 3.3 follows from Lemma 3.2.

Definition 3.4 (Totally disconnected group). A (locally compact topological Hausdorff) group G is called totally disconnected if it satisfies one of the following equivalent conditions:

(T) G is totally disconnected as a topological space, i.e. each component consists of one point;

(D) The covering dimension of the topological space G is zero;

(FS) Any element of G has a fundamental system of compact open neighborhoods.

We have to explain why these three conditions are equivalent. The implication (T) ⇒ (D) ⇒ (FS) is shown in [33, Theorem 7.7 on page 62]. It remains to prove (FS) ⇒ (T). Let U be a subset of G containing two distinct points g and h. Let V be a compact open neighborhood of x which does not contain y. Then U is the disjoint union of the open non-empty sets V ∩ U and V^{c} ∩ U and hence disconnected.

Lemma 3.5. Let G be a totally disconnected group and F a family satisfying COMOP ⊆ F ⊆ COM. Then the following square commutes up to G-homotopy and consists of G-homotopy equivalences

E_{COMOP}(G) ⟶^{u} J_{COMOP}(G)
↓  ↓
E_{F}(G) ⟶_{u} J_{F}(G)

where all maps come from the universal properties.

Proof. We first show that any compact subgroup H ⊆ G is contained in a compact open subgroup. From [33, Theorem 7.7 on page 62] we get a compact open subgroup K ⊆ G. Since H is compact, we can find finitely many elements h_{1}, h_{2}, …, h_{s} in H such that H ⊆ ⋃_{i=1}^{s} h_{i}K. Put L := ⋂_{h∈H} hKh^{−1}. Then hLh^{−1} = L for all h ∈ H. Since L = ⋂_{i=1}^{s} h_{i}Kh_{i}^{−1} holds, L is compact open. Hence LH is a compact open subgroup containing H.

This implies that J_{F}(G) is COMOP-numerable. Obviously J_{COMOP}(G) is F-numerable. We conclude from the universal properties that J_{COMOP}(G) → J_{F}(G) is a G-homotopy equivalence.

The map u: E_{COMOP}(G) → J_{COMOP}(G) is a G-homotopy equivalence because of Lemma 3.3.

This and Theorem 2.5 (iii) imply that E_{COMOP}(G)^{H} is contractible for all H ∈ F. Hence E_{COMOP}(G) → E_{F}(G) is a G-homotopy equivalence by Theorem 1.9 (ii).

Definition 3.6 (Almost connected group). Given a group G, let G^{0} be the normal subgroup given by the component of the identity and \overline{G} = G/G^{0} be the component group. We call G almost connected if its component group \overline{G} is compact.

A Lie group G is almost connected if and only if it has finitely many path components. In particular a discrete group is almost connected if it is finite.

Theorem 3.7 (Comparison of E_{F}(G) and J_{F}(G)). The map u: E_{F}(G) → J_{F}(G) defined in 3.1 is a G-homotopy equivalence if one of the following conditions is satisfied:

(i) Each element in F is an open subgroup of G;

(ii) The group G is discrete;

(iii) The group G is a Lie group and every element H ∈ F is compact;

(iv) The group G is totally disconnected and F = COM or F = COMOP;

(v) The group G is almost connected and each element in F is compact.

Proof. Assertions (i), (ii), (iii) and (iv) have already been proved in Lemma 3.3 and Lemma 3.5. Assertion (v) follows from Lemma 3.2 and Theorem 4.3.

The following example shows that the map u: E_{F}(G) → J_{F}(G) defined in 3.1 is in general not a G-homotopy equivalence.

Example 3.8 (Totally disconnected groups and TR). Let G be totally disconnected. We claim that u: E_{TR}(G) → J_{TR}(G) defined in 3.1 is a Ghomotopy equivalence if and only if G is discrete. In view of Theorem 2.5 (iii) and Lemma 3.3 this is equivalent to the statement that E_{TR}(G) is contractible if and only if G is discrete. If G is discrete, we already know that E_{TR}(G) is contractible. Suppose now that E_{TR}(G) is contractible. We obtain a numerable G-principal bundle G → E_{TR}(G) → G\E_{TR}(G) by Remark 2.8. This implies that it is a fibration by a result of Hurewicz [84, Theorem on p. 33]. Since E_{TR}(G) is contractible, G and the loop space Ω(G\E_{TR}(G)) are homotopy equivalent [84, 6.9^{∗} on p. 137, 6.10^{∗} on p. 138, Corollary 7.27 on p. 40]. Since G\E_{TR}(G) is a CW -complex, Ω(G\E_{TR}(G)) has the homotopy type of a CW - complex [62]. Hence there exists a homotopy equivalence f: G → X be from G to a CW -complex X. Then the induced map π_{0}(G) → π_{0}(X) between the set of path components is bijective. Hence the preimage of each path component of X is a path component of G and therefore a point since G is totally disconnected. Since X is locally path-connected each path component of X is open in X. We conclude that G is the disjoint union of the preimages of the path components of X and each of these preimages is open in G and consists of one point. Hence G is discrete.

Remark 3.9 (Compactly generated spaces). In the following theorem we will work in the category of compactly generated spaces. This convenient category is explained in detail in [73] and [84, I.4]. A reader may ignore this technical point in the following theorem without harm, but we nevertheless give a short explanation.

A Hausdorff space X is called compactly generated if a subset A ⊆ X is closed if and only if A ∩ K is closed for every compact subset K ⊆ X. Given a topological space X, let k(X) be the compactly generated topological space with the same underlying set as X and the topology for which a subset A ⊆ X is closed if and only if for every compact subset K ⊆ X the intersection A ∩ K is closed in the given topology on X. The identity induces a continuous map k(X) → X which is a homeomorphism if and only if X is compactly generated. The spaces X and k(X) have the same compact subsets. Locally compact Hausdorff spaces and every Hausdorff space which satisfies the first axiom of countability are compactly generated. In particular metrizable spaces are compactly generated.

Working in the category of compactly generated spaces means that one only considers compactly generated spaces and whenever a topological construction such as the cartesian product or the mapping space leads out of this category, one retopologizes the result as described above to get a compactly generated space. The advantage is for example that in the category of compactly generated spaces the exponential map map(X × Y, Z) → map(X, map(Y, Z)) is always a homeomorphism, for an identification p: X → Y the map p × id_{Z}: X × Z → Y × Z is always an identification and for a filtration by closed subspaces X_{1} ⊂ X_{2} ⊆ … ⊆ X such that X is the colimit colim_{n→∞} X_{n}, we always get X × Y = colim_{n→∞}(X_{n} × Y). In particular the product of a G-CW -complex X with a H-CW -complex Y is in a canonical way a G × H-CW -complex X × Y. Since we are assuming that G is a locally compact Hausdorff group, any G-CW - complex X is compactly generated.

The following result has grown out of discussions with Ralf Meyer.

Theorem 3.10 (Equality of \underline{E}G and \underline{J}G). Let G be a locally compact second counbtable topological Hausdorff group. Then the canonical G-map \underline{E}G → \underline{J}G is a G-homotopy equivalence.

Proof. In the sequel of the proof we work in the category of compactly generated spaces (see Remark 3.9). Notice that the model mentioned in Theorem 2.5 (i) is metrizable and hence compactly generated (see [7, Appendix 1]). Because of Lemma 3.2 it suffices to construct a G-CW -complex Z with compact isotropy groups together with a G-map \underline{J}G → Z.

Let G^{0} be the component of the identity which is a normal closed subgroup. Let p: G → G/G^{0} be the projection. The groups G^{0} and G/G^{0} are locally compact second countable Hausdorff groups and G/G^{0} is totally disconnected. We conclude from Lemma 3.5 that there is a G-map \underline{J}(G/G^{0}) → E_{COMOP}(G/G^{0}). Since \underline{J}G is COMOP-numerable, the G/G^{0}-space G^{0}\(\underline{J}G) is COM-numerable and hence there exists a G/G^{0}-map G^{0}\(\underline{J}G) → \underline{J}(G/G^{0}). Thus we get a G-map u: \underline{J}G → res_{p} E_{COMOP}(G/G^{0}), where the G-CW -complex res_{p} E_{COMOP}(G/G^{0}) is obtained from the G/G^{0}-CW -complex E_{COMOP}(G/G^{0}) by letting g ∈ G act by p(g). We obtain a G-map id ×f: \underline{J}G → \underline{J}G × res_{p} E_{COMOP}(G/G^{0}). Hence it suffices to construct a G-CW -complex Z with compact isotropy groups together with a G-map f: \underline{J}G × res_{p} E_{COMOP}(G/G^{0}) → Z. For this purpose we construct a sequence of G-CW -complexes Z_{−1} ⊆ Z_{0} ⊆ Z_{1} ⊆ … such that Z_{n} is a G-CW -subcomplex of Z_{n+1} and each Z_{n} has compact isotropy groups, and G-homotopy equivalences f_{n}: res_{p} E_{COMOP}(G/G^{0})_{n} × \underline{J}G → Z_{n}. with f_{n+1}|_{res_{p} E_{COMOP} (G/G^{0})_{n}} = f_{n}, where E_{COMOP}(G/G^{0})_{n} is the n-skeleton of E_{COMOP}(G/G^{0}). The canonical G-map

colim_{n→∞} (\underline{J}G × res_{p} E_{COMOP}(G/G^{0})_{n}) → \underline{J}G × res_{p} E_{COMOP}(G/G^{0})

is a G-homeomorphism. The G-space Z = colim_{n→∞} Z_{n} is a G-CW -complex with compact isotropy groups. Hence we can define the desired G-map by f = colim_{n→∞} f_{n} after we have constructed the G-maps f_{n}. This will be done by induction over n. The induction beginning n = −1 is given by id: ∅ → ∅. The induction step from n to (n+1) is done as follows. Choose a G/G^{0}-pushout

∐_{i∈I}(G/G^{0})/H_{i} × S^{n} ⟶ E_{COMOP}(G/G^{0})_{n}
↓  ↓
∐_{i∈I}(G/G^{0})/H_{i} × D^{n+1} ⟶ E_{COMOP}(G/G^{0})_{n+1}

where each H_{i} is a compact open subgroup of G/G^{0}. We obtain a G-pushout

∐_{i∈I} res_{p} ((G/G^{0})/H_{i} × S^{n}) × \underline{J}G ⟶ res_{p} E_{COMOP}(G/G^{0})_{n} × \underline{J}G
↓  ↓
∐_{i∈I} res_{p} ((G/G^{0})/H_{i} × D^{n+1}) × \underline{J}G ⟶ res_{p} E_{COMOP}(G/G^{0})_{n+1} × \underline{J}G

In the sequel let K_{i} ⊆ G be the open almost connected subgroup p^{−1}(H_{i}). The G-spaces res_{p}(G/G^{0})/H_{i} and G/K_{i} agree. We have the G-homeomorphism

G ×_{K_{i}} res_{G}^{K_{i}} \underline{J}G ⟶^{≅} G/K_{i} × \underline{J}G, (g, x) ↦ (gK_{i}, gx).

Thus we obtain a G-pushout

(∐_{i∈I} G ×_{K_{i}} (res_{G}^{K_{i}} \underline{J}G)) × S^{n} ⟶^{w} res_{p} E_{COMOP}(G/G^{0})_{n} × \underline{J}G
id ×i↓  ↓
(∐_{i∈I} G ×_{K_{i}} (res_{G}^{K_{i}} \underline{J}G)) × D^{n+1} ⟶ res_{p} E_{COMOP}(G/G^{0})_{n+1} × \underline{J}G

(3.11)

where i: S^{n} → D^{n+1} is the obvious inclusion.

Let X be a COM-numerable K_{i}-space. Then the G-space G ×_{K_{i}} \underline{J}K_{i} is a COM-numerable and hence admits a G-map to \underline{J}G. Its restriction to \underline{J}K_{i} = K_{i} ×_{K_{i}} \underline{J}K_{i} defines a K_{i}-map f: X → res_{G}^{K_{i}} \underline{J}G. If f_{1} and f_{2} are K_{i}-maps X → res_{G}^{K_{i}} \underline{J}G, we obtain G-maps \overline{f_{k}}: G ×_{K_{i}} X → \underline{J}G by sending (g, x) → gf_{k}(x) for k = 0, 1. By the universal property of \underline{J}G these two G-maps are G-homotopic. Hence f_{0} and f_{1} are K_{i}-homotopic. Since K_{i} ⊆ G is open, res_{G}^{K_{i}} \underline{J}G is a COMnumerable K_{i}-space. Hence the K_{i}-space res_{G}^{K_{i}} \underline{J}G is a model for \underline{J}K_{i}. Since K_{i} is almost connected, there is a K_{i}-homotopy equivalence \underline{E}K_{i} → res_{G}^{K_{i}} \underline{J}G by Theorem 3.7 (v). Hence we obtain a G-homotopy equivalence

u_{i}: G ×_{K_{i}} \underline{E}K_{i} → G ×_{K_{i}} (res_{G}^{K_{i}} \underline{J}G)

with a K_{i}-CW -complex with compact isotropy groups as source.

In the sequel we abbreviate

X_{n} := res_{p} E_{COMOP}(G/G^{0})_{n} × \underline{J}G;
Y := ∐_{i∈I} G ×_{K_{i}} (res_{G}^{K_{i}} \underline{J}G);
Y′ = ∐_{i∈I} G ×_{K_{i}} \underline{E}K_{i}.

Choose a G-homotopy equivalence v: Y′ → Y. By the equivariant cellular Approximatiom Theorem we can find a G-homotopy h: Y′ × S^{n} × [0, 1] → Z_{n} such that h_{0} = f_{n} ∘ w ∘ (v × id_{S^{n}}) and the G-map h_{1} : Y × S^{n} → Z_{n} is cellular. Consider the following commutative diagram of G-spaces

Y × D^{n+1}  ⟵^{id_{Y} ×i}  Y × S^{n}  ⟶^{w}  X_{n}
id↓  ^{id}↓  ^{f_{n}}↓
Y × D^{n+1}  ⟵^{id_{Y} ×i}  Y × S^{n}  ⟶^{f_{n}∘w}  Z_{n}
v×id_{D^{n+1}} ↑  ^{v×id_{S^{n}}} ↑  ^{id}↑
Y′ × D^{n+1}  ⟵^{id_{Y ′} ×i}  Y′ × S^{n}  ⟶^{f_{n}∘w∘(v×id_{S^{n}})} Z_{n}
j_{0}↓  ^{j_{0}}↓  ^{id}↓
Y′ × D^{n+1} × [0, 1] ⟵^{id_{Y ′} ×i×id_{[0,1]}} Y′ × S^{n} × [0, 1]  ⟶^{h}  Z_{n}
j_{1}↑  ^{j_{1}}↑  ^{id}↑
Y′ × D^{n+1}  ⟵^{id_{Y ′} ×i}  Y′ × S^{n}  ⟶^{h_{1}}  Z_{n}

where j_{0} and j_{1} always denotes the obvious inclusions. The G-pushout of the top row is X_{n+1} by (3.11). Let Z_{n+1} be the G-pushout of the bottom row. This is a G-CW -complex with compact isotropy groups containing Z_{n} as G-CW - subcomplex. Let W_{2} and W_{3} and W_{4} be the G-pushout of the second, third and fourth row. The diagram above induces a sequence of G-maps

X_{n+1} ⟶^{u_{1}} W_{2} ⟵^{u_{2}} W_{3} ⟶^{u_{3}} W_{4} ⟵^{u_{4}} Z_{n+1}

The left horizontal arrow in each row is a G-cofibration as i is a cofibration. Each of the vertical arrows is a G-homotopy equivalence. This implies that each of the maps u_{1}, u_{2}, u_{3} and u_{4} are G-homotopy equivalences. Notice that we can consider Z_{n} as a subspace of W_{2}, W_{3}, W_{4} such that the inclusion Z_{n} → W_{k} is a G-cofibration. Each of the maps u_{2}, u_{3} and u_{4} induces the identity on Z_{n}, whereas u_{1} induces f_{n} on X_{n}. By a cofibration argument one can find Ghomotopy inverses u_{2}^{−1} and u_{4}^{−1} of u_{2} and u_{4} which induce the identity on Z_{n}. Now define the desired G-homotopy equivalence f_{n+1} : X_{n+1} → Z_{n+1} to be the composition u_{4}^{−1} ∘ u_{3} ∘ u_{2}^{−1} ∘ u_{1}. This finishes the proof of Theorem 3.10.

4 Special Models

In this section we present some interesting geometric models for the space E_{F}(G) and J_{F}(G) focussing on \underline{E}G and \underline{J}G. In particular we are interested in cases, where these models satisfy finiteness conditions such as being finite, finite dimensional or of finite type.

One extreme case is, where we take F to be the family ALL of all subgroups. Then a model for both E_{ALL}(G) and J_{ALL}(G) is G/G. The other extreme case is the family TR consisting of the trivial subgroup. This case has already been treated in Example 2.7, Remark 2.8 and Example 3.8.

4.1 Operator Theoretic Model

Let G be a locally compact Hausdorff topological group. Let C_{0}(G) be the Banach space of complex valued functions of G vanishing at infinity with the supremum-norm. The group G acts isometrically on C_{0}(G) by (g · f)(x) := f(g^{−1}x) for f ∈ C_{0}(G) and g, x ∈ G. Let P C_{0}(G) be the subspace of C_{0}(G) consisting of functions f such that f is not identically zero and has non-negative real numbers as values.

The next theorem is due to Abels [1, Theorem 2.4].

Theorem 4.1 (Operator theoretic model). The G-space P C_{0}(G) is a model for \underline{J}G.

Remark 4.2. Let G be discrete. Another model for \underline{J}G is the space

X_{G} = {f: G → [0, 1] | f has finite support, ∑_{g∈G} f(g) = 1}

with the topology coming from the supremum norm [7, page 248]. Let P_{∞}(G) be the geometric realization of the simplicial set whose k-simplices consist of (k + 1)-tupels (g_{0}, g_{1}, …, g_{k}) of elements g_{i} in G. This also a model for \underline{E}G [1, Example 2.6]. The spaces X_{G} and P_{∞}(G) have the same underlying sets but in general they have different topologies. The identity map induces a (continuous) G-map P_{∞}(G) → X_{G} which is a G-homotopy equivalence, but in general not a G-homeomorphism (see also [80, A.2]).

4.2 Almost Connected Groups

The next result is due to Abels [1, Corollary 4.14].

Theorem 4.3 (Almost connected groups). Let G be a (locally compact Hausdorff) topological group. Suppose that G is almost connected, i.e. the group G/G^{0} is compact for G^{0} the component of the identity element. Then G contains a maximal compact subgroup K which is unique up to conjugation. The G-space G/K is a model for \underline{J}G.

The next result follows from Example 1.4, Theorem 3.7 (iii) and Theorem 4.3.

Theorem 4.4 (Discrete subgroups of almost connected Lie groups). Let L be a Lie group with finitely many path components. Then L contains a maximal compact subgroup K which is unique up to conjugation. The L-space L/K is a model for \underline{E}L.

If G ⊆ L is a discrete subgroup of L, then L/K with the obvious left G-action is a finite dimensional G-CW -model for \underline{E}G.

4.3 Actions on Simply Connected Non-Positively Curved Manifolds

The next theorem is due to Abels [1, Theorem 4.15].

Theorem 4.5 (Actions on simply connected non-positively curved manifolds). Let G be a (locally compact Hausdorff) topological group. Suppose that G acts properly and isometrically on the simply-connected complete Riemannian manifold M with non-positive sectional curvature. Then M is a model for \underline{J}G.

4.4 Actions on CAT(0)-spaces

Theorem 4.6 (Actions on CAT(0)-spaces). Let G be a (locally compact Hausdorff) topological group. Let X be a proper G-CW -complex. Suppose that X has the structure of a complete CAT(0)-space for which G acts by isometries. Then X is a model for \underline{E}G.

Proof. By [13, Corollary II.2.8 on page 179] the K-fixed point set of X is a nonempty convex subset of X and hence contractible for any compact subgroup K ⊂ G.

This result contains as special case Theorem 4.5 and partially Theorem 4.7 since simply-connected complete Riemannian manifolds with non-positive sectional curvature and trees are CAT(0)-spaces.

4.5 Actions on Trees and Graphs of Groups

A tree is a 1-dimensional CW -complex which is contractible.

Theorem 4.7 (Actions on trees). Suppose that G acts continuously on a tree T such that for each element g ∈ G and each open cell e with g · e ∩ e ≠ ∅ we have gx = x for any x ∈ e. Assume that the isotropy group of each x ∈ T is compact.

Then G can be written as an extension 1 → K → G → \overline{G} → 1 of a compact group containing G^{0} and a totally disconnected group \overline{G} such that K acts trivially and T is a 1-dimensional model for

E_{COM}(G) = J_{COM}(G) = E_{COMOP}(G) = J_{COMOP}(G).

Proof. We conclude from Remark 1.3 that T is a G-CW -complex and all isotropy groups are compact open. Let K be the intersection of all the isotropy groups of points of T. This is a normal compact subgroup of G which contains the component of the identity G^{0}. Put \overline{G} = G/K. This is a totally disconnected group. Let H ⊆ G be compact. If e_{0} is a zero-cell in T, then H · e_{0} is a compact discrete set and hence finite. Let T′ be the union of all geodesics with extremities in H · e. This is a H-invariant subtree of T of finite diameter. One shows now inductively over the diameter of T′ that T′ has a vertex which is fixed under the H-action (see [69, page 20] or [25, Proposition 4.7 on page 17]). Hence T^{H} is non-empty. If e and f are vertices in T^{H}, the geodesic in T from e to f must be H-invariant. Hence T^{H} is a connected CW -subcomplex of the tree T and hence is itself a tree. This shows that T^{H} is contractible. Hence T is a model for E_{COM}(G) = E_{COM}(\overline{G}). Now apply Lemma 3.5.

Let G be a locally compact Hausdorff group. Suppose that G acts continuously on a tree T such that for each element g ∈ G and each open cell e with g · e ∩ e ≠ ∅ we have gx = x for any x ∈ e. If the G-action on a tree has possibly not compact isotropy groups, one can nevertheless get nice models for E_{COMOP}(G) as follows. Let V be the set of equivariant 0-cells and E be the set of equivariant 1-cells of T. Then we can choose a G-pushout

\begin{matrix}∐_{e∈E} G/H_{e} × \{−1, 1\} & ⟶^{q} & T_{0} = ∐_{v∈V} G/K_{v} \\ ↓ & & ↓ \\ ∐_{e∈E} G/H_{e} × [−1, 1] & ⟶ & T \end{matrix}  (4.8)

where the left vertical arrow is the obvious inclusion. Fix e ∈ E and σ ∈ {−1, 1}. Choose elements v(e, σ) ∈ V and g(e, σ) ∈ G such that q restricted to G/H_{e} × {σ} is the G-map G/H_{e} → G/K_{v(e,σ)} which sends 1H_{e} to g(e, σ)K_{v(e,σ)}. Then conjugation with g(e, σ) induces a group homomorphism c_{g(e,σ)} : H_{e} → K_{v(e,±1)} and there is an up to equivariant homotopy unique c_{g(e,σ)}-equivariant cellular map f_{g(e,σ)} : E_{COMOP}(H_{e}) → E_{COMOP}(K_{e(g,σ)}). Define a G-map

Q: ∐_{e∈E} G ×_{H_{e}} E_{COMOP}(H_{e}) × {−1, 1} → ∐_{v∈V} G ×_{K_{v}} E_{COMOP}(K_{v})

by requiring that the restriction of Q to G×_{H_{e}} E_{COMOP}(H_{e})×{σ} is the G-map

G ×_{H_{e}} E_{COMOP}(H_{e}) → G ×_{K_{v(e,σ)}} E_{COMOP}(K_{g(e,σ)}), (g, x) ↦ (g, f_{g(e,σ)}(x)).

Let T_{COMOP} be the G-pushout

∐_{e∈E} G ×_{H_{e}} E_{COMOP}(H_{e}) × {−1, 1} ⟶^{Q} ∐_{v∈V} G ×_{K_{v}} E_{COMOP}(K_{v})
↓  ↓
∐_{e∈E} G ×_{H_{e}} E_{COMOP}(H_{e}) × [−1, 1] ⟶  T_{COMOP}

The G-space T_{COMOP} inherits a canonical G-CW -structure with compact open isotropy groups. Notice that for any open subgroup L ⊆ G one can choose as model for E_{COMOP}(L) the restriction res_{G}^{L} E_{COMOP}(G) of E_{COMOP}(G) to L and that there is a G-homeomorphism G×_{L} res_{G}^{L} E_{COMOP}(G) ⟶^{≅} G/L×E_{COMOP}(G) which sends (g, x) to (gL, gx). This implies that T_{COMOP} is G-homotopy equivalent to T ×E_{COMOP}(G) with the diagonal G-action. If H ⊆ G is compact open, then T^{H} is contractible. Hence (T × E_{COMOP}(G))^{H} is contractible for compact open subgroup H ⊆ G. Theorem 1.9 (ii) shows

Theorem 4.9 (Models based on actions on trees). The G-CW -complex T_{COMOP} is a model for E_{COMOP}(G).

The point is that it may be possible to choose nice models for the various spaces E_{COMOP}(H_{e}) and E_{COMOP}(K_{v}) and thus get a nice model for E_{COMOP}(G). If all isotropy groups of the G-action on T are compact, we can choose all spaces E_{COMOP}(H_{e}) and E_{COMOP}(K_{v}) to be {pt.} and we rediscover Theorem 4.7.

Next we recall which discrete groups G act on trees. Recall that an oriented graph X is a 1-dimensional CW -complex together with an orientation for each 1-cell. This can be codified by specifying a triple (V, E, s: E × {−1, 1} → V) consisting of two sets V and E and a map s. The associated oriented graph is the pushout

E × {−1, 1} ⟶^{s} V
↓  ↓
E × [0, 1] ⟶ X

So V is the set of vertices, E the set of edges, and for a edge e ∈ E its initial vertex is s(e, −1) and its terminal vertex is s(e, 1). A graph of groups G on a connected oriented graph X consists of two sets of groups {K_{v} | v ∈ V } and {H_{e} | e ∈ E} with V and E as index sets together with injective group homomorphisms ϕ_{v,σ}: H_{e} → K_{s(e,σ)} for each e ∈ E. Let X_{0} ⊆ X be some maximal tree. We can associate to these data the fundamental group π = π(G, X, X_{0}) as follows. Generators of π are the elements in K_{v} for each v ∈ V and the set {t_{e} | e ∈ E}. The relations are the relations in each group K_{v} for each v ∈ V, the relation t_{e} = 1 for e ∈ V if e belongs to X_{0}, and for each e ∈ E and h ∈ H_{e} we require t_{e}^{−1}ϕ_{e,−1}(h)t_{e} = ϕ_{e,+1}(h). It turns out that the obvious map K_{v} → π is an injective group homomorphism for each v ∈ V and we will identify in the sequel K_{v} with its image in π [25, Corollary 7.5 on page 33], [69, Corollary 1 in 5.2 on page 45]. We can assign to these data a tree T = T (X, X_{0}, G) with π-action as follows. Define a π-map

q: ∐_{e∈E} π/ im(ϕ_{e,−1}) × {−1, 1} → ∐_{v∈V} π/K_{v}

by requiring that its restriction to π/ im(ϕ_{e,−1})×{−1} is the π-map given by the projection π/ im(ϕ_{e,−1}) → π/K_{s(e,−1)} and its restriction to π/ im(ϕ_{e,−1}) × {1} is the π-map π/ im(ϕ_{e,−1}) → π/K_{s(e,1)} which sends g im(ϕ_{e,−1}) to gt_{e} im(ϕ_{e,1}). Now define a 1-dimensional G-CW -complex T = T (G, X, X_{0}) using this π-map q and the π-pushout analogous to (4.8). It turns out that T is contractible [25, Theorem 7.6 on page 33], [69, Theorem 12 in 5.3 on page 52].

On the other hand, suppose that T is a 1-dimensional G-CW -complex. Choose a G-pushout (4.8). Let X be the connected oriented graph G\T. It has a set of vertices V and as set of edges the set E. The required map s: E × {−1, 1} → V sends s(e, σ) to the vertex for which q(G/H_{e} × {σ}) meets and hence is equal to G/K_{s(e,σ)}. Moreover, we get a graph of groups G on X as follows. Let {K_{v} | v ∈ V } and {H_{e} | e ∈ E} be the set of groups given by (4.8). Choose an element g(e, σ) ∈ G such that the G-map induced by q from G/H_{e} to G/K_{s(e,σ)} sends 1H_{e} to g(e, σ)K_{s(e,σ)}. Then conjugation with g(e, σ) induces a group homomorphism ϕ_{e,σ}: H_{e} → K_{s(e,σ)}. After a choice of a maximal tree X_{0} in X one obtains an isomorphism G ≅ π(G, X, X_{0}). (Up to isomorphism) we get a bijective correspondence between pairs (G, T) consisting of a group G acting on an oriented tree T and a graph of groups on connected oriented graphs. For details we refer for instance to [25, I.4 and I.7] and [69, §5].

Example 4.10 (The graph associated to amalgamated products). Consider the graph D with one edge e and two vertices v_{−1} and v_{1} and the map s: {e} × {−1, 1} → {v_{−1}, v_{1}} which sends (e, σ) to v_{σ}. Of course this is just the graph consisting of a single segment which is homeomorphic to [−1, 1]. Let G be a graph of groups on D. This is the same as specifying a group H_{e} and groups K_{−1} and K_{1} together with injective group homomorphisms ϕ_{σ}: H_{e} → K_{σ} for σ ∈ {−1, 1}. There is only one choice of a maximal subtree in D, namely D itself. Then the fundamental group π of this graph of groups is the amalgamated product of K_{−1} and K_{1} over H_{e} with respect to ϕ_{−1} and ϕ_{1}, i.e. the pushout of groups

H_{e} ⟶^{ϕ_{−1}} K_{−1}
ϕ_{1}↓  ↓
K_{1} ⟶ π

Choose ϕ_{σ}-equivariant maps f_{σ}: \underline{E}H_{e} → \underline{E}K_{σ}. They induce π-maps

F_{σ}: π ×_{H_{e}} \underline{E}H_{e} → π ×_{K_{σ}} \underline{E}K_{σ}, (g, x) ↦ (g, f_{σ}(x)).

We get a model for \underline{E}π as the π-pushout

π ×_{H_{e}} \underline{E}H_{e} × \{−1, 1\} \underrightarrow{F_{−1} ∐ F_{1}} π ×_{K_{−1}} \underline{E}K_{−1} ∐ π ×_{K_{1}} \underline{E}K_{1}
↓  ↓
π ×_{H_{e}} \underline{E}H_{e} × [−1, 1] ⟶  \underline{E}π

Example 4.11 (The graph associated to an HNN-extension). Consider the graph S with one edge e and one vertex v. There is only one choice for the map s: {e} × {−1, 1} → {v}. Of course this graph is homeomorphic to S^{1}. Let G be a graph of groups on S. It consists of two groups H_{e} and K_{v} and two injective group homomorphisms ϕ_{σ}: H_{e} → K_{v} for σ ∈ {−1, 1}. There is only one choice of a maximal subtree, namely {v}. The fundamental group π of G is the so called HNN-extension associated to the data ϕ_{σ}: H_{e} → K_{v} for σ ∈ {−1, 1}, i.e. the group generated by the elements of K_{v} and a letter t_{v} whose relations are those of K_{v} and the relations t_{v}^{−1}ϕ_{−1}(h)t_{v} = ϕ_{1}(h) for all h ∈ H_{e}. Recall that the natural map K_{v} → π is injective and we will identify K_{v} with its image in π. Choose ϕ_{σ}-equivariant maps f_{σ}: \underline{E}H_{e} → \underline{E}K_{v}. Let F_{σ}: π ×_{ϕ_{−1}} \underline{E}H_{e} → π × \underline{E}K_{v} be the π-map which sends (g, x) to gf_{−1}(x) for σ = −1 and to gt_{e}f_{1}(x) for σ = 1. Then a model for \underline{E}π is given by the π-pushout

π ×_{ϕ_{−1}} \underline{E}H_{e} × \{−1, 1\} \underrightarrow{F_{−1} ∐ F_{1}} π ×_{K_{v}} \underline{E}K_{v}
↓  ↓
π ×_{ϕ_{−1}} \underline{E}H_{e} × [−1, 1] ⟶  \underline{E}π

Notice that this looks like a telescope construction which is infinite to both sides. Consider the special case, where H_{e} = K_{v}, ϕ_{−1} = id and ϕ_{1} is an automorphism. Then π is the semidirect product K_{v} ⋊_{ϕ_{1}} ℤ. Choose a ϕ_{1}- equivariant map f_{1} : \underline{E}K_{v} → \underline{E}K_{v}. Then a model for \underline{E}π is given by the to both side infinite mapping telescope of f_{1} with the K_{v} ⋊_{ϕ_{1}} ℤ action, for which ℤ acts by shifting to the right and k ∈ K_{v} acts on the part belonging to n ∈ ℤ by multiplication with ϕ_{1}^{n}(k). If we additionally assume that ϕ_{1} = id, then π = K_{v} × ℤ and we get \underline{E}K_{v} × ℝ as model for \underline{E}π.

Remark 4.12. All these constructions yield also models for EG = E_{TR}(G) if one replaces everywhere the spaces \underline{E}H_{e} and \underline{E}K_{v} by the spaces EH_{e} and EK_{v}.

4.6 Affine Buildings

Let Σ be an affine building, sometimes also called Euclidean building. This is a simplicial complex together with a system of subcomplexes called apartments satisfying the following axioms:

(i) Each apartment is isomorphic to an affine Coxeter complex; (ii) Any two simplices of Σ are contained in some common apartment;

(iii) If two apartments both contain two simplices A and B of Σ, then there is an isomorphism of one apartment onto the other which fixes the two simplices A and B pointwise.

The precise definition of an affine Coxeter complex, which is sometimes called also Euclidean Coxeter complex, can be found in [17, Section 2 in Chapter VI], where also more information about affine buildings is given. An affine building comes with metric d: Σ × Σ → [0, ∞) which is non-positively curved and complete. The building with this metric is a CAT(0)-space. A simplicial automorphism of Σ is always an isometry with respect to d. For two points x, y in the affine building there is a unique line segment [x, y] joining x and y. It is the set of points {z ∈ Σ | d(x, y) = d(x, z) + d(z, y)}. For x, y ∈ Σ and t ∈ [0, 1] let tx + (1 − t)y be the point z ∈ Σ uniquely determined by the property that d(x, z) = td(x, y) and d(z, y) = (1 − t)d(x, y). Then the map

r: Σ × Σ × [0, 1] → Σ, (x, y, t) ↦ tx + (1 − t)y

is continuous. This implies that Σ is contractible. All these facts are taken from [17, Section 3 in Chapter VI] and [13, Theorem 10A.4 on page 344].

Suppose that the group G acts on Σ by isometries. If G maps a non-empty bounded subset A of Σ to itself, then the G-action has a fixed point [17, Theorem 1 in Section 4 in Chapter VI on page 157]. Moreover the G-fixed point set must be contractible since for two points x, y ∈ Σ^{G} also the segment [x, y] must lie in Σ^{G} and hence the map r above induces a continuous map Σ^{G} × Σ^{G} × [0, 1] → Σ^{G}. This implies together with Theorem 1.9 (ii), Example 1.5, Lemma 3.3 and Lemma 3.5

Theorem 4.13 (Affine buildings). Let G be a topological (locally compact Hausdorff group). Suppose that G acts on the affine building by simplicial automorphisms such that each isotropy group is compact. Then each isotropy group is compact open, Σ is a model for J_{COMOP}(G) and the barycentric subdivision Σ′ is a model for both J_{COMOP}(G) and E_{COMOP}(G). If we additionally assume that G is totally disconnected, then Σ is a model for both \underline{J}G and \underline{E}G.

Example 4.14 (Bruhat-Tits building). An important example is the case of a reductive p-adic algebraic group G and its associated affine Bruhat-Tits building β(G) [75],[76]. Then β(G) is a model for \underline{J}G and β(G)′ is a model for \underline{E}G by Theorem 4.13.

4.7 The Rips Complex of a Word-Hyperbolic Group

A metric space X = (X, d) is called δ-hyperbolic for a given real number δ ≥ 0 if for any four points x, y, z, t the following inequality holds

d(x, y) + d(z, t) ≤ max{d(x, z) + d(y, t), d(x, t) + d(y, z)} + 2δ. (4.15)

A group G with a finite symmetric set S of generators is called δ-hyperbolic if the metric space (G, d_{S}) given by G and the wordlwength metric with respect to one ( and hence all) finite symmetric set of generators is δ-hyperbolic.

The Rips complex P_{d}(G, S) of a group G with a symmetric finite set S of generators for a natural number d is the geometric realization of the simplicial set whose set of k-simplices consists of (k + 1)-tuples (g_{0}, g_{1}, … g_{k}) of pairwise distinct elements g_{i} ∈ G satisfying d_{S}(g_{i}, g_{j}) ≤ d for all i, j ∈ {0, 1, …, k}. The obvious G-action by simplicial automorphisms on P_{d}(G, S) induces a Gaction by simplicial automorphisms on the barycentric subdivision P_{d}(G, S)′ (see Example 1.5). The following result is proved in [60], [61].

Theorem 4.16 (Rips complex). Let G be a (discrete) group with a finite symmetric set of generators. Suppose that (G, S) is δ-hyperbolic for the real number δ ≥ 0. Let d be a natural number with d ≥ 16δ +8. Then the barycentric subdivision of the Rips complex P_{d}(G, S)′ is a finite G-CW -model for \underline{E}G.

A metric space is called hyperbolic if it is δ-hyperbolic for some real number δ ≥ 0. A finitely generated group G is called hyperbolic if for one (and hence all) finite symmetric set S of generators the metric space (G, d_{S}) is a hyperbolic metric space. Since for metric spaces the property hyperbolic is invariant under quasiisometry and for two symmetric finite sets S_{1} and S_{2} of generators of G the metric spaces (G, d_{S_{1}}) and (G, d_{S_{2}}) are quasiisometric, the choice of S does not matter. Theorem 4.16 implies that for a hyperbolic group there is a finite G-CW -model for \underline{E}G.

The notion of a hyperbolic group is due to Gromov and has intensively been studied (see for example [13], [29], [30]). The prototype is the fundamental group of a closed hyperbolic manifold.

4.8 Arithmetic Groups

Arithmetic groups in a semisimple connected linear ℚ-algebraic group possess finite models for \underline{E}G. Namely, let G(ℝ) be the ℝ-points of a semisimple ℚgroup G(ℚ) and let K ⊆ G(ℝ) a maximal compact subgroup. If A ⊆ G(ℚ) is an arithmetic group, then G(ℝ)/K with the left A-action is a model for E_{FIN} (A) as already explained in Theorem 4.4. The A-space G(ℝ)/K is not necessarily cocompact. The Borel-Serre completion of G(ℝ)/K (see [10], [68]) is a finite A-CW -model for E_{FIN} (A) as pointed out in [2, Remark 5.8], where a private communication with Borel and Prasad is mentioned.

4.9 Outer Automorphism Groups of Free groups

Let F_{n} be the free group of rank n. Denote by Out(F_{n}) the group of outer automorphisms of F_{n}, i.e. the quotient of the group of all automorphisms of F_{n} by the normal subgroup of inner automorphisms. Culler and Vogtmann [21],

[81] have constructed a space X_{n} called outer space on which Out(F_{n}) acts with finite isotropy groups. It is analogous to the Teichmüller space of a surface with the action of the mapping class group of the surface. Fix a graph R_{n} with one vertex v and n-edges and identify F_{n} with π_{1}(R_{n}, v). A marked metric graph (g, Γ) consists of a graph Γ with all vertices of valence at least three, a homotopy equivalence g: R_{n} → Γ called marking and to every edge of Γ there is assigned a positive length which makes Γ into a metric space by the path metric. We call two marked metric graphs (g, Γ) and (g′, Γ′) equivalent of there is a homothety h: Γ → Γ′ such that g ∘ h and h′ are homotopic. Homothety means that there is a constant λ > 0 with d(h(x), h(y)) = λ · d(x, y) for all x, y. Elements in outer space X_{n} are equivalence classes of marked graphs. The main result in [21] is that X is contractible. Actually, for each finite subgroup H ⊆ Out(F_{n}) the H-fixed point set X_{n}^{H} is contractible [44, Propostion 3.3 and Theorem 8.1], [83, Theorem 5.1].

The space X_{n} contains a spine K_{n} which is an Out(F_{n})-equivariant deformation retraction. This space K_{n} is a simplicial complex of dimension (2n − 3) on which the Out(F_{n})-action is by simplicial automorphisms and cocompact. Actually the group of simplicial automorphisms of K_{n} is Out(F_{n}) [14]. Hence the barycentric subdivision K′_{n} is a finite (2n − 3)-dimensional model of \underline{E} Out(F_{n}).

4.10 Mapping Class groups

Let Γ_{g,r}^{s} be the mapping class group of an orientable compact surface F of genus g with s punctures and r boundary components. This is the group of isotopy classes of orientation preserving selfdiffeomorphisms F_{g} → F_{g}, which preserve the punctures individually and restrict to the identity on the boundary. We require that the isotopies leave the boundary pointwise fixed. We will always assume that 2g + s + r > 2, or, equivalently, that the Euler characteristic of the punctured surface F is negative. It is well-known that the associated Teichmüller space T_{g,r}^{s} is a contractible space on which Γ_{g,r}^{s} acts properly. Actually T_{g,r}^{s} is a model for E_{FIN} (Γ_{g,r}^{s}) by the results of Kerckhoff [42].

We could not find a clear reference in the literature for the to experts known statement that there exist a finite Γ_{g,r}^{s}-CW -model for E_{FIN} (Γ_{g,r}^{s}). The work of Harer [32] on the existence of a spine and the construction of the spaces T_{S}(ϵ)^{H} due to Ivanov [37, Theorem 5.4.A] seem to lead to such models.

4.11 Groups with Appropriate Maximal Finite Subgroups

Let G be a discrete group. Let MFIN be the subset of FIN consisting of elements in FIN which are maximal in FIN. Consider the following assertions concerning G:

(M) Every non-trivial finite subgroup of G is contained in a unique maximal finite subgroup;

(NM) M ∈ MFIN , M ≠ {1} ⇒ N_{G}M = M;

For such a group there is a nice model for \underline{E}G with as few non-free cells as possible. Let {(M_{i}) | i ∈ I} be the set of conjugacy classes of maximal finite subgroups of M_{i} ⊆ Q. By attaching free G-cells we get an inclusion of G-CW - complexes j_{1} : ∐_{i∈I} G ×_{M_{i}} EM_{i} → EG, where EG is the same as E_{TR}(G), i.e. a contractible free G-CW -complex. Define \underline{E}G as the G-pushout

\begin{matrix}∐_{i∈I} G ×_{M_{i}} EM_{i} & ⟶^{j_{1}} & EG \\ ^{u_{1}}↓ & & ↓^{f_{1}} \\ ∐_{i∈I} G/M_{i} & ⟶_{k_{1}} & \underline{E}G \end{matrix}, EG  (4.17)

where u_{1} is the obvious G-map obtained by collapsing each EM_{i} to a point.

We have to explain why \underline{E}G is a model for the classifying space for proper actions of G. Obviously it is a G-CW -complex. Its isotropy groups are all finite. We have to show for H ⊆ G finite that (\underline{E}G)^{H} contractible. We begin with the case H ≠ {1}. Because of conditions (M) and (NM) there is precisely one index i_{0} ∈ I such that H is subconjugated to M_{i_{0}} and is not subconjugated to M_{i} for i ≠ i_{0} and we get

(∐_{i∈I} G/M_{i})^{H} = (G/M_{i_{0}})^{H} = {pt.}.

Hence \underline{E}G^{H} = \{pt.\}. It remains to treat H = \{1\}. Since u_{1} is a non-equivariant homotopy equivalence and j_{1} is a cofibration, f_{1} is a non-equivariant homotopy equivalence and hence \underline{E}G is contractible (after forgetting the group action).

Here are some examples of groups Q which satisfy conditions (M) and (NM):

• Extensions 1 → ℤ^{n} → G → F → 1 for finite F such that the conjugation action of F on ℤ^{n} is free outside 0 ∈ ℤ^{n}. The conditions (M), (NM) are satisfied by [58, Lemma 6.3].

• Fuchsian groups F The conditions (M), (NM) are satisfied (see for instance [58, Lemma 4.5]). In [58] the larger class of cocompact planar groups (sometimes also called cocompact NEC-groups) is treated.

• One-relator groups G Let G be a one-relator group. Let G = ⟨(q_{i})_{i∈I} | r⟩ be a presentation with one relation. We only have to consider the case, where G contains torsion. Let F be the free group with basis {q_{i} | i ∈ I}. Then r is an element in F. There exists an element s ∈ F and an integer m ≥ 2 such that r = s^{m}, the cyclic subgroup C generated by the class \overline{s} ∈ G represented by s has order m, any finite subgroup of G is subconjugated to C and for any g ∈ G the implication g^{−1}Cg ∩ C ≠ 1 ⇒ g ∈ C holds. These claims follows from [59, Propositions 5.17, 5.18 and 5.19 in II.5 on pages 107 and 108]. Hence G satisfies (M) and (NM).

4.12 One-Relator Groups

Let G be a one-relator group. Let G = ⟨(q_{i})_{i∈I} | r⟩ be a presentation with one relation. There is up to conjugacy one maximal finite subgroup C which is cyclic. Let p: ∗_{i∈I} ℤ → G be the epimorphism from the free group generated by the set I to G, which sends the generator i ∈ I to q_{i}. Let Y → ⋁_{i∈I} S^{1} be the G-covering associated to the epimorphism p. There is a 1-dimensional unitary C-representation V and a C-map f: SV → res_{G}^{C} Y such that the following is true. The induced action on the unit sphere SV is free. If we equip SV and DV with the obvious C-CW -complex structures, the C-map f can be chosen to be cellular and we obtain a G-CW -model for \underline{E}G by the G-pushout

G ×_{C} SV ⟶^{\overline{f}} Y
↓  ↓
G ×_{C} DV ⟶ \underline{E}G

where \overline{f} sends (g, x) to gf(x). Thus we get a 2-dimensional G-CW -model for \underline{E}G such that \underline{E}G is obtained from G/C for a maximal finite cyclic subgroup C ⊆ G by attaching free cells of dimensions ≤ 2 and the CW -complex structure on the quotient G\\underline{E}G has precisely one 0-cell, precisely one 2-cell and as many 1-cells as there are elements in I. All these claims follow from [16, Exercise 2 (c) II. 5 on page 44].

If G is torsionfree, the 2-dimensional complex associated to a presentation with one relation is a model for BG (see also [59, Chapter III §§9 -11]).

4.13 Special Linear Groups of (2,2)-Matrices

In order to illustrate some of the general statements above we consider the special example SL_{2}(ℝ) and SL_{2}(ℤ).

Let ℍ^{2} be the 2-dimensional hyperbolic space. We will use either the upper half-plane model or the Poincaré disk model. The group SL_{2}(ℝ) acts by isometric diffeomorphisms on the upper half-plane by Moebius transformations, i.e. a

matrix ( \begin{matrix}a & b \\ c & d \end{matrix} ) acts by sending a complex number z with positive imaginary

part to \frac{az+b}{cz+d}. This action is proper and transitive. The isotropy group of z = i is SO(2). Since ℍ^{2} is a simply-connected Riemannian manifold, whose sectional curvature is constant −1, the SL_{2}(ℝ)-space ℍ^{2} is a model for \underline{E}SL_{2}(ℝ) by Theorem 4.5.

One easily checks that SL_{2}(ℝ) is a connected Lie group and SO(2) ⊆ SL_{2}(ℝ) is a maximal compact subgroup. Hence SL_{2}(ℝ)/SO(2) is a model for \underline{E}SL_{2}(ℝ) by Theorem 4.3. Since the SL_{2}(ℝ)-action on ℍ^{2} is transitive and SO(2) is the isotropy group at i ∈ ℍ^{2}, we see that the SL_{2}(ℝ)-manifolds SL_{2}(ℝ)/SO(2) and ℍ^{2} are SL_{2}(ℝ)-diffeomorphic.

Since SL_{2}(ℤ) is a discrete subgroup of SL_{2}(ℝ), the space ℍ^{2} with the obvious SL_{2}(ℤ)-action is a model for \underline{E}SL_{2}(ℤ) (see Theorem 4.4).

The group SL_{2}(ℤ) is isomorphic to the amalgamated product ℤ/4 ∗_{ℤ/2} ℤ/6. From Example 4.10 we conclude that a model for \underline{E}SL_{2}(ℤ) is given by the following SL_{2}(ℤ)-pushout

SL_{2}(ℤ)/(ℤ/2) × \{−1, 1\} \underrightarrow{F_{−1} ∐ F_{1}} SL_{2}(ℤ)/(ℤ/4) ∐ SL_{2}(ℤ)/(ℤ/6)
↓  ↓
SL_{2}(ℤ)/(ℤ/2) × [−1, 1] ⟶  \underline{E}SL_{2}(ℤ)

where F_{−1} and F_{1} are the obvious projections. This model for \underline{E}SL_{2}(ℤ) is a tree, which has alternately two and three edges emanating from each vertex. The other model ℍ^{2} is a manifold. These two models must be SL_{2}(ℤ)-homotopy equivalent. They can explicitly be related by the following construction.

Divide the Poincaré disk into fundamental domains for the SL_{2}(ℤ)-action. Each fundamental domain is a geodesic triangle with one vertex at infinity, i.e. a vertex on the boundary sphere, and two vertices in the interior. Then the union of the edges, whose end points lie in the interior of the Poincaré disk, is a tree T with SL_{2}(ℤ)-action. This is the tree model above. The tree is a SL_{2}(ℤ)-equivariant deformation retraction of the Poincaré disk. A retraction is given by moving a point p in the Poincaré disk along a geodesic starting at the vertex at infinity, which belongs to the triangle containing p, through p to the first intersection point of this geodesic with T.

The tree T above can be identified with the Bruhat-Tits building of SL_{2}(ℚ_{p}) and hence is a model for \underline{E}SL_{2}(ℚ_{p}) (see [17, page 134]). Since SL_{2}(ℤ) is a discrete subgroup of SL_{2}(ℚ_{p}), we get another reason why this tree is a model for SL_{2}(ℤ).

4.14 Manifold Models

It is an interesting question, whether one can find a model for \underline{E}G which is a smooth G-manifold. One may also ask whether such a manifold model realizes the minimal dimension for \underline{E}G or whether the action is cocompact. Theorem 4.5 gives some information about these questions for simply connected nonpositively curved Riemannian manifolds and Theorem 5.24 for discrete subgroups of Lie groups with finitely many path components. On the other hand there exists a virtually torsionfree group G such that G acts properly and cocompactly on a contractible manifold (without boundary), but there is no finite G-CW -model for \underline{E}G [23, Theorem 1.1].

5 Finiteness Conditions

In this section we investigate whether there are models for E_{F}(G) which satisfy certain finiteness conditions such as being finite, being of finite type or being of finite dimension as a G-CW -complex.

5.1 Review of Finiteness Conditions on BG

As an illustration we review the corresponding question for EG for a discrete group G. This is equivalent to the question whether for a given discrete group G there is a CW -complex model for BG which is finite, of finite type or finite dimensional.

We introduce the following notation. Let R be a commutative associative ring with unit. The trivial RG-module is R viewed as RG-module by the trivial G-action. A projective resolution or free resolution respectively for an RGmodule M is an RG-chain complex P_{∗} of projective or free respectively RGmodules with P_{i} = 0 for i ≤ −1 such that H_{i}(P_{∗}) = 0 for i ≥ 1 and H_{0}(P_{∗}) is RG-isomorphic to M. If additionally each RG-module P_{i} is finitely generated and P_{∗} is finite dimensional, we call P_{∗} finite.

An RG-module M has cohomological dimension cd(M) ≤ n, if there exists a projective resolution of dimension ≤ n for M. This is equivalent to the condition that for any RG-module N we have Ext_{RG}^{i}(M, N) = 0 for i ≥ n + 1. A group G has cohomological dimension cd(G) ≤ n over R if the trivial RG-module R has cohomological dimension ≤ n. An RG-module M is of type FP_{n}, if it admits a projective RG-resolution P_{∗} such that P_{i} is finitely generated for i ≤ n and of type FP_{∞} if it admits a projective RG-resolution P_{∗} such that P_{i} is finitely generated for all i. A group G is of type FP_{n} or FP_{∞} respectively if the trivial ℤG-module ℤ is of type FP_{n} or FP_{∞} respectively.

Here is a summary of well-known statements about finiteness conditions on BG. A key ingredient in the proof of the next result is the fact that the cellular RG-chain complex C_{∗}(EG) is a free and in particular a projective RGresolution of the trivial RG-module R since EG is a free G-CW -complex and contractible, and that C_{∗}(EG) is n-dimensional or of type FP_{n} respectively if BG is n-dimensional or has finite n-skeleton respectively.

Theorem 5.1 (Finiteness conditions for BG). Let G be a discrete group.

(i) If there exists a finite dimensional model for BG, then G is torsionfree;

(ii) (a) There exists a CW -model for BG with finite 1-skeleton if and only if G is finitely generated;

(b) There exists a CW -model for BG with finite 2-skeleton if and only if G is finitely presented;

(c) For n ≥ 3 there exists a CW -model for BG with finite n-skeleton if and only if G is finitely presented and of type FP_{n};

(d) There exists a CW -model for BG of finite type, i.e. all skeleta are finite, if and only if G is finitely presented and of type FP_{∞};

(e) There exists groups G which are of type FP_{2} and which are not finitely presented;

(iii) There is a finite CW -model for BG if and only if G is finitely presented and there is a finite free ℤG-resolution F_{∗} for the trivial ℤG-module ℤ; (iv) The following assertions are equivalent:

(a) The cohomological dimension of G is ≤ 1;

(b) There is a model for BG of dimension ≤ 1;

(c) G is free.

(v) The following assertions are equivalent for d ≥ 3:

(a) There exists a CW -model for BG of dimension ≤ d;

(b) G has cohomological dimension ≤ d over ℤ;

(vi) For Thompson’s group F there is a CW -model of finite type for BG but no finite dimensional model for BG.

Proof. (i) Suppose we can choose a finite dimensional model for BG. Let C ⊆ G be a finite cyclic subgroup. Then C\B̃G = C\EG is a finite dimensional model for BC. Hence there is an integer d such that we have H_{i}(BC) = 0 for i ≥ d. This implies that C is trivial [16, (2.1) in II.3 on page 35]. Hence G is torsionfree. (ii) See [9] and [16, Theorem 7.1 in VIII.7 on page 205]. (iii) See [16, Theorem 7.1 in VIII.7 on page 205]. (iv) See [71] and [74].

(v) See [16, Theorem 7.1 in VIII.7 on page 205]. (vi) See [18].

5.2 Modules over the Orbit Category

Let G be a discrete group and let F be a family of subgroups. The orbit category Or(G) of G is the small category, whose objects are homogeneous G-spaces G/H and whose morphisms are G-maps. Let Or_{F}(G) be the full subcategory of Or(G) consisting of those objects G/H for which H belongs to F. A ℤOr_{F}(G)-module is a contravariant functor from Or_{F}(G) to the category of ℤ-modules. A morphism of such modules is a natural transformation. The category of ℤOr_{F}(G)-modules inherits the structure of an abelian category from the standard structure of an abelian category on the category of ℤ-modules. In particular the notion of a projective ℤOr_{F}(G)-module is defined. The free ℤOr_{F}(G)-module ℤ map(G/?, G/K) based at the object G/K is the ℤOr_{F}(G)- module that assigns to an object G/H the free ℤ-module ℤ map_{G}(G/H, G/K) generated by the set map_{G}(G/H, G/K). The key property of it is that for any ℤOr_{F}(G)-module N there is a natural bijection of ℤ-modules

hom_{ℤOr _{F} (G)}(ℤ map_{G}(G/?, G/K), N) ⟶^{≅} N(G/K), ϕ ↦ ϕ(G/K)(id_{G/K}).

This is a direct consequence of the Yoneda Lemma. A ℤOr_{F}(G)-module is free if it is isomorphic to a direct sum ⨁_{i∈I} ℤ map(G/?, G/K_{i}) for appropriate choice of objects G/K_{i} and index set I. A ℤOr_{F}(G)-module is called finitely generated if it is a quotient of a ℤOr_{F}(G)-module of the shape ⨁_{i∈I} ℤ map(G/?, G/K_{i}) with a finite index set I. Notice that a lot of standard facts for ℤ-modules carry over to ℤOr_{F}(G)-modules. For instance, a ℤOr_{F}(G)-module is projective or finitely generated projective respectively if and only if it is a direct summand in a free ℤOr_{F}(G)-module or a finitely generated free ℤOr_{F}(G)-module respectively. The notion of a projective resolution P_{∗} of a ℤOr_{F}(G)-module is obvious and notions like of cohomological dimension ≤ n or of type FP_{∞} carry directly over. Each ℤOr_{F}(G)-module has a projective resolution. The trivial ℤOr_{F}(G)- module \underline{ℤ} is the constant functor from Or_{F}(G) to the category of ℤ-modules, which sends any morphism to id: ℤ → ℤ. More information about modules over a category can be found for instance in [48, Section 9].

The next result is proved in [54, Theorem 0.1]. A key ingredient in the proof of the next result is the fact that the cellular ROr_{F}(G)-chain complex C_{∗}(E_{F}(G)) is a free and in particular a projective ROr_{F}(G)-resolution of the trivial ROr_{F}(G)-module R.

Theorem 5.2 (Algebraic and geometric finiteness conditions). Let G be a discrete group and let d ≥ 3. Then we have:

(i) There is G-CW -model of dimension ≤ d for E_{F}(G) if and only if the trivial ℤOr_{F}(G)-module \underline{ℤ} has cohomological dimension ≤ d;

(ii) There is a G-CW -model for E_{F}(G) of finite type if and only if E_{F}(G) has a G-CW -model with finite 2-skeleton and the trivial ℤOr_{F}(G)-module \underline{ℤ} is of type FP_{∞};

(iii) There is a finite G-CW -model for E_{F}(G) if and only if E_{F}(G) has a G-CW -model with finite 2-skeleton and the trivial ℤOr_{F}(G)-module \underline{ℤ} has a finite free resolution over Or_{F}(G);

(iv) There is a G-CW -model with finite 2-skeleton for \underline{E}G = E_{FIN} (G) if and only if there are only finitely many conjugacy classes of finite subgroups H ⊂ G and for any finite subgroup H ⊂ G its Weyl group W_{G}H := N_{G}H/H is finitely presented.

In the case, where we take F to be the trivial family, Theorem 5.2 (i) reduces to Theorem 5.1 (v), Theorem 5.2 (ii) to Theorem 5.1 (ii)d and Theorem 5.2 (iii) to Theorem 5.1 (iii), and one should compare Theorem 5.2 (iv) to Theorem 5.1 (ii)b.

Remark 5.3. Nucinkis [67] investigates the notion of FIN -cohomological dimension and relates it to the question whether there are finite dimensional modules for \underline{E}G. It gives another lower bound for the dimension of a model for \underline{E}G but is not sharp in general [11].

5.3 Reduction from Topological Groups to Discrete Groups

The discretization G_{d} of a topological group G is the same group but now with the discrete topology. Given a family F of (closed) subgroups of G, denote by F_{d} the same set of subgroups, but now in connection with G_{d}. Notice that F_{d} is again a family. We will need the following condition

(S)  For any closed subgroup H ⊂ G the projection p: G → G/H has a local cross section, i.e. there is a neighborhood U of eH together with a map s: U → G satisfying p ∘ s = id_{U}.

Condition (S) is automatically satisfied if G is discrete, if G is a Lie group, or more generally, if G is locally compact and second countable and has finite covering dimension [65]. The metric needed in [65] follows under our assumptions, since a locally compact Hausdorff space is regular and regularity in a second countable space implies metrizability.

The following two results are proved in [54, Theorem 0.2 and Theorem 0.3].

Theorem 5.4 (Passage from totally disconnected groups to discrete groups). Let G be a locally compact totally disconnected Hausdorff group and let F be a family of subgroups of G. Then there is a G-CW -model for E_{F}(G) that is d-dimensional or finite or of finite type respectively if and only if there is a G_{d}-CW -model for E_{F_{d}}(G_{d}) that is d-dimensional or finite or of finite type respectively.

Theorem 5.5 (Passage from topological groups to totally disconnected groups). Let G be a locally compact Hausdorff group satisfying condition (S). Put \overline{G} := G/G^{0}. Then there is a G-CW -model for \underline{E}G that is d-dimensional or finite or of finite type respectively if and only if \underline{E}\overline{G} has a \overline{G}-CW -model that is d-dimensional or finite or of finite type respectively.

If we combine Theorem 5.2, Theorem 5.4 and Theorem 5.5 we get

Theorem 5.6 (Passage from topological groups to discrete groups). Let G be a locally compact group satisfying (S). Denote by \overline{COM} the family of compact subgroups of its component group \overline{G} and let d ≥ 3. Then

(i) There is a d-dimensional G-CW -model for \underline{E}G if and only if the trivial ℤOr_{\overline{COM}_{d}}(\overline{G}_{d})-module \underline{ℤ} has cohomological dimension ≤ d;

(ii) There is a G-CW -model for \underline{G} of finite type if and only if E_{\overline{COM}_{d}}(\overline{G}_{d}) has a \overline{G}_{d}-CW -model with finite 2-skeleton and the trivial ℤOr_{\overline{COM}_{d}}(\overline{G}_{d})-module \underline{ℤ} is of type FP_{∞};

(iii) There is a finite G-CW -model for \underline{E}G if and only if E_{\overline{COM}_{d}}(\overline{G}_{d}) has a \overline{G}_{d}-CW -model with finite 2-skeleton and the trivial ℤOr_{\overline{COM}_{d}}(\overline{G}_{d})-module \underline{ℤ} has a finite free resolution.

In particular we see from Theorem 5.5 that, for a Lie group G, type questions about \underline{E}G are equivalent to the corresponding type questions of \underline{E}π_{0}(G), since π_{0}(G) = \overline{G} is discrete. In this case the family \overline{COM}_{d} appearing in Theorem 5.6. is just the family FIN of finite subgroups of π_{0}(G).

5.4 Poset of Finite Subgroups

Throughout this Subsection 5.4 let G be a discrete group. Define the G-poset

P(G) := {K | K ⊂ G finite, K ≠ 1}.  (5.7)

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

Let N_{G}H be the normalizer and let W_{G}H := N_{G}H/H be the Weyl group of H ⊂ G. Notice for a G-space X that X^{H} inherits a W_{G}H-action. Denote by CX the cone over X. Notice that C∅ is the one-point-space.

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

Lemma 5.8. The W_{G}H-space \underline{E}G^{H} is a W_{G}H-CW -model for \underline{E}W_{G}H. In particular, if \underline{E}G has a G-CW -model which is finite, of finite type or d-dimensional respectively, then there is a W_{G}H-model for \underline{E}W_{G}H which is finite, of finite type or d-dimensional respectively.

Notation 5.9 (The condition b(d) and B(d)). Let d ≥ 0 be an integer. A group G satisfies the condition b(d) or b(<∞) respectively if any ℤG-module M with the property that M restricted to ℤK is projective for all finite subgroups K ⊂ G has a projective ℤG-resolution of dimension d or of finite dimension respectively. A group G satisfies the condition B(d) if W_{G}H satisfies the condition b(d) for any finite subgroup H ⊂ G.

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 5.10. Suppose that there is a d-dimensional G-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 G-CW -model for \underline{E}G. Then G satisfies condition B(d).

Proof. Let H ⊂ G be finite. Then X/H satisfies H_{p}(X/H; ℤ) = H_{p}(∗, ℤ) for all p ≥ 0 [12, III.5.4 on page 131]. Let C_{∗} be the cellular ℤW_{G}H-chain complex of X/H. This is a d-dimensional resolution of the trivial ℤW_{G}H-module ℤ and each chain module is a sum of ℤW_{G}H-modules of the shape ℤ[W_{G}H/K] for some finite subgroup K ⊂ W_{G}H. Let N be a ℤW_{G}H-module such that N is projective over ℤK for any finite subgroup K ⊂ W_{G}H. Then C_{∗} ⊗_{ℤ} N with the diagonal W_{G}H-operation is a d-dimensional projective ℤW_{G}H-resolution of N.

Theorem 5.11 (An algebraic criterion for finite dimensionality). Let G be a discrete group. Suppose that we have for any finite subgroup H ⊂ G an integer d(H) ≥ 3 such that d(H) ≥ d(K) for H ⊂ K and d(H) = d(K) if H and K are conjugate in G. Consider the following statements:

(i) There is a G-CW -model \underline{E}G such that for any finite subgroup H ⊂ G

dim(\underline{E}G^{H}) = d(H);

(ii) We have for any finite subgroup H ⊂ G and for any ℤW_{G}H-module M

H_{ℤW_{G}H}^{d(H)+1}(EW_{G}H × (C|P(W_{G}H)|, |P(W_{G}H)|); M) = 0;

(iii) We have for any finite subgroup H ⊂ G that its Weyl group W_{G}H satisfies b(< ∞) and that there is a subgroup Δ(H) ⊂ W_{G}H of finite index such that for any ℤΔ(H)-module M

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

Then (i) implies both (ii) and (iii). If there is an upper bound on the length l(H) of the finite subgroups H of G, then these statements (i), (ii) and (iii) are equivalent.

The proof of Theorem 5.11 can be found in [49, Theorem 1.6]. In the case that G has finite virtual cohomological dimension a similar result is proved in [20, Theorem III].

Example 5.12. Suppose that G is torsionfree. Then Theorem 5.11 reduces to the well-known result [16, 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:

(i) There is a d-dimensional CW -model for BG;

(ii) G has cohomological dimension ≤ d;

(iii) G has virtual cohomological dimension ≤ d.

Remark 5.13. If W_{G}H contains a non-trivial normal finite subgroup L, then |P(W_{G}H)| is contractible and

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

The proof of this fact is given in [49, Example 1.8].

The next result is taken from [49, Theorem 1.10]. A weaker version of it for certain classes of groups and in l exponential dimension estimate can be found in [43, Theorem B] (see [49, Remark 1.12]).

Theorem 5.14 (An upper bound on the dimension). Let G be a group and let l ≥ 0 and d ≥ 0 be integers such that the length l(H) of any finite subgroup H ⊂ G is bounded by l and G satisfies B(d). Then there is a G-CW -model for \underline{E}G such that for any finite subgroup H ⊂ G

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

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

5.5 Extensions of Groups

In this subsection we consider an exact sequence of discrete groups 1 → Δ → G → π → 1. 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 G. The proof of the next Theorem 5.15 is taken from [49, Theorem 3.1]), the proof of Theorem 5.16 is an easy variation.

Theorem 5.15 (Dimension bounds and extensions). 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}G has a (dk + m)-dimensional G-CW - model.

Theorem 5.16. Suppose that Δ has the property that for any group Γ which contains Δ as subgroup of finite index, there is a k-dimensional Γ-CW -model for \underline{E}Γ. Suppose that \underline{E}π has a m-dimensional π-CW -model. Then \underline{E}G has a (k + m)-dimensional G-CW -model.

We will see in Example 5.26 that the condition about Δ in Theorem 5.16 is automatically satisfied if Δ is virtually poly-cyclic.

The next two results are taken from [49, Theorem 3.2 and Theorem 3.3]).

Theorem 5.17. 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}G has a finite G-CW -model or a G-CW -model of finite type respectively.

Theorem 5.18. 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}G has a finite G-CW -model or a G-CW -model of finite type respectively.

5.6 One-Dimensional Models for \underline{E}G

The following result follows from Dunwoody [27, Theorem 1.1].

Theorem 5.19 (A criterion for 1-dimensional models). Let G be a discrete group. Then there exists a 1-dimensional model for \underline{E}G if and only the cohomological dimension of G over the rationals ℚ is less or equal to one.

If G is finitely generated, then there is a 1-dimensional model for \underline{E}G if and only if G contains a finitely generated free subgroup of finite index [41, Theorem 1]. If G is torsionfree, we rediscover the results due to Swan and Stallings stated in Theorem 5.1 (iv) from Theorem 5.19.

5.7 Groups of Finite Virtual Dimension

In this section we investigate the condition b(d) and B(d) of Notation 5.9 for a discrete group G and explain how our results specialize in the case of a group of finite virtual cohomological dimension.

Remark 5.20. There exists groups G with a finite dimensional model for \underline{E}G, which do not admit a torsionfree subgroup of finite index. For instance, let G be a countable locally finite group which is not finite. Then its cohomological dimension over the rationals is ≤ 1 and hence it possesses a 1-dimensional model for \underline{E}G by Theorem 5.19. Obviously it contains no torsionfree subgroup of finite index. An example of a group G with a finite 2-dimensional model for \underline{E}G, which does not admit a torsionfree subgroup of finite index, is described in [11, page 493].

A discrete group G has virtual cohomological dimension ≤ d if and only if it contains a torsionfree subgroup Δ of finite index such that Δ has cohomological dimension ≤ d. This is independent of the choice of Δ ⊆ G because for two torsionfree subgroups Δ, Δ′ ⊆ G we have that Δ has cohomological dimension ≤ d if and only if Δ′ has cohomological dimension ≤ d. The next two results are taken from [49, Lemma 6.1, Theorem 6.3, Theorem 6.4].

Lemma 5.21. If G satisfies b(d) or B(d) respectively, then any subgroup Δ of G satisfies b(d) or B(d) respectively.

Theorem 5.22 (Virtual cohomological dimension and the condition B(d)). If G contains a torsionfree subgroup Δ of finite index, then the following assertions are equivalent:

(i) G satisfies B(d);

(ii) G satisfies b(d);

(iii) G has virtual cohomological dimension ≤ d.

Next we improve Theorem 5.14 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 ⊂ G.

Theorem 5.23 (Virtual cohomological dimension and dim(\underline{E}G). Let G be a discrete group which contains a torsionfree subgroup of finite index and has virtual cohomological dimension vcd(G) ≤ d. Let l ≥ 0 be an integer such that the length l(H) of any finite subgroup H ⊂ G is bounded by l.

Then we have vcd(G) ≤ dim(\underline{E}G) for any model for \underline{E}G and there is a G-CW -model for \underline{E}G such that for any finite subgroup H ⊂ G

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

holds. In particular there exists a model for \underline{E}G of dimension max\{3, d\} + l.

Theorem 5.24 (Discrete subgroups of Lie groups). Let L be a Lie group with finitely many path components. Then L contains a maximal compact subgroup K which is unique up to conjugation. Let G ⊆ L be a discrete subgroup of L. Then L/K with the left G-action is a model for \underline{E}G.

Suppose additionally that G contains a torsionfree subgroup Δ ⊆ G of finite index. Then we have

vcd(G) ≤ dim(L/K)

and equality holds if and only if G\L is compact.

Proof. We have already mentioned in Theorem 4.4 that L/K is a model for \underline{E}G. The restriction of \underline{E}G to Δ is a Δ-CW -model for \underline{E}Δ and hence Δ\\underline{E}G is a CW -model for BΔ. This implies vcd(G) := cd(Δ) ≤ dim(L/K). Obviously Δ\L/K is a manifold without boundary. Suppose that Δ\L/K is compact. Then Δ\L/K is a closed manifold and hence its homology with ℤ/2-coefficients in the top dimension is non-trivial. This implies cd(Δ) ≥ dim(Δ\L/K) and hence vcd(G) = dim(L/K). If Δ\L/K is not compact, it contains a CW - complex X ⊆ Δ\L/K of dimension smaller than Δ\L/K such that the inclusion of X into Δ\L/K is a homotopy equivalence. Hence X is another model for BΔ. This implies cd(Δ) < dim(L/K) and hence vcd(G) < dim(L/K).

Remark 5.25. An often useful strategy to find smaller models for E_{F}(G) is to look for a G-CW -subcomplex X ⊆ E_{F}(G) such that there exists a G-retraction r: E_{F}(G) → X, i.e. a G-map r with r|_{X} = id_{X}. Then X is automatically another model for E_{F}(G). We have seen this already in the case SL_{2}(ℤ), where we found a tree inside ℍ^{2} = SL_{2}(ℝ)/SO(2) as explained in Subsection 4.13. This method can be used to construct a model for \underline{E}SL_{n}(ℤ) of dimension \frac{n(n−1)}{2} and to show that the virtual cohomological dimension of SL_{n}(ℤ) is \frac{n(n−1)}{2}. Notice that SL_{n}(ℝ)/SO(n) is also a model for \underline{E}SL_{n}(ℤ) by Theorem 4.4 but has dimension \frac{n(n+1)}{2} − 1.

Example 5.26 (Virtually poly-cyclic groups). Let the group Δ be virtually poly-cyclic, i.e. Δ contains a subgroup Δ′ of finite index for which there is a finite sequence {1} = Δ′_{0} ⊆ Δ′_{1} ⊆ … ⊆ Δ′_{n} = Δ′ of subgroups such that Δ′_{i−1} is normal in Δ′_{i} with cyclic quotient Δ′_{i}/Δ′_{i−1} for i = 1, 2, …, n. Denote by r the number of elements i ∈ {1, 2, …, n} with Δ′_{i}/Δ′_{i−1} ≅ ℤ. The number r is called the Hirsch rank. The group Δ contains a torsionfree subgroup of finite index. We call Δ′ poly-ℤ if r = n, i.e. all quotients Δ′_{i}/Δ′_{i−1} are infinite cyclic. We want to show:

(i) r = vcd(Δ);

(ii) r = max{i | H_{i}(Δ′; ℤ/2) ≠ 0} for one (and hence all) poly-ℤ subgroup Δ′ ⊂ Δ of finite index;

(iii) There exists a finite r-dimensional model for \underline{E}Δ and for any model \underline{E}Δ we have dim(\underline{E}Δ) ≥ r.

We use induction over the number r. If r = 0, then Δ is finite and all the claims are obviously true. Next we explain the induction step from (r − 1) to r ≥ 1. We can choose an extension 1 → Δ_{0} → Δ ⟶^{p} V → 1 for some virtually poly-cyclic group Δ_{0} with r(Δ_{0}) = r(Δ) − 1 and some group V which contains ℤ as subgroup of finite index. The induction hypothesis applies to any group Γ which contains Δ_{0} as subgroup of finite index. Since V maps surjectively to ℤ or the infinite dihedral group D_{∞} with finite kernel and both ℤ and D_{∞} have 1-dimensional models for their classifying space for proper group actions, there is a 1-dimensional model for \underline{E}V. We conclude from Theorem 5.16 that there is a r-dimensional model for \underline{E}Δ.

The existence of a r-dimensional model for \underline{E}Δ implies vcd(Δ) ≤ r.

For any torsionfree subgroup Δ′ ⊂ Δ of finite index we have max{i | H_{i}(Δ′; ℤ/2) ≠ 0} ≤ vcd(Δ),

It is not hard to check by induction over r that we can find a sequence of torsionfree subgroups {1} ⊆ Δ_{0} ⊆ Δ_{1} ⊆ … ⊆ Δ_{r} ⊆ Δ such that Δ_{i−1} is normal in Δ_{i} with Δ_{i}/Δ_{i−1} ≅ ℤ for i ∈ {1, 2, …, r} and Δ_{r} has finite index in Δ. We show by induction over i that H_{i}(Δ_{i}; ℤ/2) = ℤ/2 for i = 0, 1, …, r. The induction beginning i = 0 is trivial. The induction step from (i − 1) to i follows from the part of the long exact Wang sequence

H_{i}(Δ_{i−1}; ℤ/2) = 0 → H_{i}(Δ_{i}; ℤ/2) → H_{i−1}(Δ_{i−1}; ℤ/2) = ℤ/2

⟶^{id −H_{i−1}(f;ℤ/2) = 0} H_{i−1}(Δ_{i−1}; ℤ/2)

which comes from the Hochschild-Serre spectral sequence associated to the extension 1 → Δ_{i−1} → Δ_{i} → ℤ → 1 for f: Δ_{i−1} → Δ_{i−1} the automorphism induced by conjugation with some preimage in Δ_{i} of the generator of ℤ. This implies

r = max{i | H_{i}(Δ_{r}; ℤ/2) ≠ 0} = cd(Δ_{r}) = vcd(Δ).

Now the claim follows.

The existence of a r-dimensional model for \underline{E}G is proved for finitely generated nilpotent groups with vcd(G) ≤ r for r ≠ 2 in [66], where also not necessarily finitely generated nilpotent groups are studied.

The work of Dekimpe-Igodt [24] or Wilking [85, Theorem 3] implies that there is a model for E_{FIN} (Δ) whose underlying space is ℝ^{r}.

5.8 Counterexamples

The following problem is stated by Brown [15, page 32]. It created a lot of activities and many of the results stated above were motivated by it.

Problem 5.27. For which discrete groups G, which contain a torsionfree subgroup of finite index and has virtual cohomological dimension ≤ d, does there exist a d-dimensional G-CW -model for \underline{E}G?

The following four problems for discrete groups G are stated in the problem lists appearing in [49] and [82].

Problem 5.28. Let H ⊆ G be a subgroup of finite index. Suppose that \underline{E}H has a H-CW -model of finite type or a finite H-CW -model respectively. Does then \underline{E}G have a G-CW -model of finite type or a finite G-CW -model respectively?

Problem 5.29. If the group G contains a subgroup of finite index H which has a H-CW -model of finite type for \underline{E}H, does then G contain only finitely many conjugacy classes of finite subgroups?

Problem 5.30. Let G be a group such that BG has a model of finite type. Is then BW_{G}H of finite type for any finite subgroup H ⊂ G?

Problem 5.31. Let 1 → Δ ⟶^{i} G ⟶^{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 G-CW -model of finite type for \underline{E}G. Is then there a π-CW -model of finite type for \underline{E}π?

Leary and Nucinkis [47] have constructed many very interesting examples of discrete groups some of which are listed below. Their main technical input is an equivariant version of the constructions due to Bestvina and Brady [9]. These examples show that the answer to the Problems 5.27, 5.28, 5.29, 5.30 and 5.31 above is not positive in general. A group G is of type VF if it contains a subgroup H ⊆ G of finite index for which there is a finite model for BH.

(i) For any positive integer d there exist a group G of type VF which has virtually cohomological dimension ≤ 3d, but for which any model for \underline{E}G has dimension ≥ 4d;

(ii) There exists a group G with a finite cyclic subgroup H ⊆ G such that G is of type VF but the centralizer C_{G}H of H in G is not of type FP_{∞};

(iii) There exists a group G of type VF which contains infinitely many conjugacy classes of finite subgroups;

(iv) There exists an extension 1 → Δ → G → π → 1 such that \underline{E}Δ and \underline{E}G have finite G-CW -models but there is no G-CW -model for \underline{E}π of finite type.

6 The Orbit Space of \underline{E}G

We will see that in many computations of the group (co-)homology, of the algebraic K- and L-theory of the group ring or the topological K-theory of the reduced C^{∗}-algebra of a discrete group G a key problem is to determine the homotopy type of the quotient space G\\underline{E}G of \underline{E}G. The following result shows that this is a difficult problem in general and can only be solved in special cases. It was proved by Leary and Nucinkis [46] based on ideas due to Baumslag-Dyer-Heller [8] and Kan and Thurston [40].

Theorem 6.1 (The homotopy type of G\\underline{E}G). Let X be a connected CW - complex. Then there exists a group G such that G\\underline{E}G is homotopy equivalent to X.

There are some cases, where the quotient G\\underline{E}G has been determined explicitly using geometric input. We mention a few examples.

(i) Let G be a planar group (sometimes also called NEC) group, i.e. a discontinuous group of isometries of the two-sphere S^{2}, the Euclidean plane ℝ^{2}, or the hyperbolic plane ℍ^{2}. Examples are Fuchsian groups and twodimensional crystallographic groups. If G acts on ℝ^{2} or ℍ^{2} and the action is cocompact, then ℝ^{2} or ℍ^{2} is a model for \underline{E}G and the quotient space G\\underline{E}G is a compact 2-dimensional surface. The number of boundary components, its genus and the answer to the question, whether G\\underline{E}G is orientable, can be read off from an explicit presentation of G. A summary of these details can be found in [58, Section 4], where further references to papers containing proofs of the stated facts are given;

(ii) Let G = ⟨(q_{i})_{i∈I} | r⟩ be a one-relator group. Let F be the free group on the letters {q_{i} | i ∈ I}. Then r is an element in F. There exists an element s ∈ F and an integer m ≥ 1 such that r = s^{m}, the cyclic subgroup C generated by the class \overline{s} ∈ G represented by s has order m, any finite subgroup of G is subconjugated to C and for any g ∈ G the implication g^{−1}Cg ∩C ≠ {1} ⇒ g ∈ C holds (see [59, Propositions 5.17, 5.18 and 5.19 in II.5 on pages 107 and 108]). In the sequel we use the two-dimensional model for \underline{E}G described in Subsection 4.12. Let us compute the integral homology of BG and G\\underline{E}G. Since G\\underline{E}G has precisely one 2-cell and is two-dimensional, H_{2}(G\\underline{E}G) is either trivial or infinite cyclic and H_{k}(G\\underline{E}G) = 0 for k ≥ 3. We obtain the short exact sequence

0 → H_{2}(BG) ⟶^{H_{2}(q)} H_{2}(G\\underline{E}G) ⟶^{∂_{2}} H_{1}(BC) ⟶^{H_{1}(Bi)} H_{1}(BG)

⟶^{H_{1}(q)} H_{1}(G\\underline{E}G) → 0

and for k ≥ 3 isomorphisms

H_{k}(Bi): H_{k}(BC) ⟶^{≅} H_{k}(BG)

from the pushout coming from (4.17)

BC ⟶^{i} BG
↓  ↓
\{pt.\} ⟶ G\\underline{E}G

Hence H_{2}(G\\underline{E}G) = 0 and the sequence

0 → H_{1}(BC) ⟶^{H_{1}(Bi)} H_{1}(BG) ⟶^{H_{1}(q)} H_{1}(G\\underline{E}G) → 0

is exact, provided that H_{2}(BG) = 0. Suppose that H_{2}(BG) ≠ 0. Hopf’s Theorem says that H_{2}(BG) ≅ R ∩ [F, F]/[F, R] if R is the subgroup of G normally generated by r ∈ F (see [16, Theorem 5.3 in II.5 on page 42]). For every element in R ∩ [F, F]/[F, R] there exists n ∈ ℤ such that r^{n} belongs to [F, F] and the element is represented by r^{n}. Hence there is n ≥ 1 such that r^{n} does belong to [F, F]. Since F/[F, F] is torsionfree, also s and r belong to [F, F]. We conclude that both H_{2}(BG) and H_{2}(G\\underline{E}G) are infinite cyclic groups, H_{1}(BC) → H_{1}(BG) is trivial and H_{1}(q): H_{1}(BG) ⟶^{≅}H_{1}(G\\underline{E}G) is bijective. We also see that H_{2}(BG) = 0 if and only if r does not belong to [F, F].

(iii) Let Hei be the three-dimensional discrete Heisenberg group which is the subgroup of GL_{3}(ℤ) consisting of upper triangular matrices with 1 on the diagonals. Consider the ℤ/4-action given by

( \begin{matrix}1 & x & y \\ 0 & 1 & z \\ 0 & 0 & 1 \end{matrix} ) ↦ ( \begin{matrix}1 & −z & y − xz \\ 0 & 1 & x \\ 0 & 0 & 1 \end{matrix} ) .

Then a key result in [53] is that G\\underline{E}G is homeomorphic to S^{3} for G = Hei ⋊ℤ/4;

(iv) A key result in [70, Corollary on page 8] implies that for G = SL_{3}(ℤ) the quotient space G\\underline{E}G is contractible.

7 Relevance and Applications of Classifying Spaces for Families

In this section we discuss some theoretical aspects which involve and rely on the notion of a classifying space for a family of subgroups.

7.1 Baum-Connes Conjecture

Let G be a locally compact second countable Hausdorff group. Using the equivariant KK-theory due to Kasparov one can assign to a COM-numerable G-space X its equivariant K-theory K_{n}^{G}(X). Let C_{r}^{∗}(G) be the reduced group C^{∗}-algebra associated to G. The goal of the Baum-Connes Conjecture is to compute the topological K-theory K_{p}(C_{r}^{∗}(G)). The following formulation is taken from [7, Conjecture 3.15].

Conjecture 7.1 (Baum-Connes Conjecture). The assembly map defined by taking the equivariant index

asmb: K_{n}^{G}(\underline{J}G) ⟶^{≅} K_{n}(C_{r}^{∗}(G))

is bijective for all n ∈ ℤ.

More information about this conjecture and its relation and application to other conjectures and problems can be found for instance in [7], [34], [57], [63], [80].

7.2 Farrell-Jones Conjecture

Let G be a discrete group. Let R be a associative ring with unit. One can construct a G-homology theory H_{∗}^{G}(X; K) graded over the integers and defined for G-CW -complexes X such that for any subgroup H ⊆ G the abelian group H_{n}^{G}(G/H; K) is isomorphic to the algebraic K-groups K_{n}(RH) for n ∈ ℤ. If R comes with an involution of rings, one can also construct a G-homology theory H_{∗}^{G}(X; L^{⟨−∞⟩}) graded over the integers and defined for G-CW -complexes X such that for any subgroup H ⊆ G the abelian group H_{n}^{G}(G/H; L^{⟨−∞⟩}) is isomorphic to the algebraic L-groups L_{n}^{−∞}(RH) for n ∈ ℤ. Let VCYC be the family of virtually cyclic subgroups of G. The goal of the Farrell-Jones Conjecture is to compute the algebraic K-groups K_{n}(RH) and the algebraic L-groups L_{n}^{−∞}(RG). The following formulation is equivalent to the original one appearing in [28, 1.6 on page 257].

Conjecture 7.2 (Farrell-Jones Conjecture). The assembly maps induced by the projection E_{VCYC}(G) → G/G

asmb: H_{n}^{G}(E_{VCYC}(G), K) → H_{n}^{G}(G/G, K) = K_{n}(RG);  (7.3)
asmb: H_{n}^{G}(E_{VCYC}(G), L^{−∞}) → H_{n}^{G}(G/G, L^{−∞}) = L_{n}^{−∞}(RG), (7.4)

are bijective for all n ∈ ℤ.

More information about this conjecture and its relation and application to other conjectures and problems can be found for instance in [28] and [57].

We mention that for a discrete group G one can formulate the Baum-Connes Conjecture in a similar fashion. Namely, one can also construct a G-homology theory H_{∗}^{G}(X; K^{top}) graded over the integers and defined for G-CW -complexes X such that for any subgroup H ⊆ G the abelian group H_{n}^{G}(G/H; K^{top}) is isomorphic to the topological K-groups K_{n}(C_{r}^{∗}(H)) for n ∈ ℤ and the assembly map appearing in the Baum-Connes Conjecture can be identified with the map induced by the projection \underline{J}G = \underline{E}G → G/G (see [22], [31]). If the ring R is regular and contains ℚ as subring, then one can replace in the Farrell-Jones Conjecture 7.2 E_{VCYC}(G) by \underline{E}G but this is not possible for arbitrary rings such as R = ℤ. This comes from the appearance of Nil-terms in the Bass-Heller-Swan decomposition which do not occur in the context of the topological K-theory of reduced C^{∗}-algebras.

Both the Baum-Connes Conjecture 7.1 and the Farrell-Jones Conjecture 7.2 allow to reduce the computation of certain K-and L-groups of the group ring or the reduced C^{∗}-algebra of a group G to the computation of certain G-homology theories applied to \underline{J}G, \underline{E}G or E_{VCYC}(G). Hence it is important to find good models for these spaces or to make predictions about their dimension or whether they are finite or of finite type.

7.3 Completion Theorem

Let G be a discrete group. For a proper finite G-CW -complex let K_{G}^{∗}(X) be its equivariant K-theory defined in terms of equivariant finite dimensional complex vector bundles over X (see [56, Theorem 3.2]). It is a G-cohomology theory with a multiplicative structure. Assume that \underline{E}G has a finite G-CW - model. Let I ⊆ K_{G}^{0}(\underline{E}G) be the augmentation ideal, i.e. the kernel of the map K^{0}(\underline{E}G) → ℤ sending the class of an equivariant complex vector bundle to its complex dimension. Let K_{G}^{∗}(\underline{E}G)_{I} be the I-adic completion of K_{G}^{∗}(\underline{E}G) and let K^{∗}(BG) be the topological K-theory of BG.

Theorem 7.5 (Completion Theorem for discrete groups). Let G be a discrete group such that there exists a finite model for \underline{E}G. Then there is a canonical isomorphism

K^{∗}(BG) ⟶^{≅} K_{G}^{∗}(\underline{E}G)_{I}.

This result is proved in [56, Theorem 4.4], where a more general statement is given provided that there is a finite dimensional model for \underline{E}G and an upper bound on the orders of finite subgroups of G. In the case where G is finite, Theorem 7.5 reduces to the Completion Theorem due to Atiyah and Segal [3], [4]. A Cocompletion Theorem for equivariant K-homology will appear in [38].

7.4 Classifying Spaces for Equivariant Bundles

In [55] the equivariant K-theory for finite proper G-CW -complexes appearing in Subsection 7.3 above is extended to arbitrary proper G-CW -complexes (including the multiplicative structure) using Γ-spaces in the sense of Segal and involving classifying spaces for equivariant vector bundles. These classifying spaces for equivariant vector bundles are again classifying spaces of certain Lie groups and certain families (see [78, Section 8 and 9 in Chapter I], [56, Lemma 2.4]).

7.5 Equivariant Homology and Cohomology

Classifying spaces for families play a role in computations of equivariant homology and cohomology for compact Lie groups such as equivariant bordism as explained in [77, Chapter 7], [78, Chapter III]. Rational computations of equivariant (co-)-homology groups are possible in general using Chern characters for discrete groups and proper G-CW -complexes (see [50], [51], [52]).

8 Computations using Classifying Spaces for Families

In this section we discuss some computations which involve and rely on the notion of a classifying space for a family of subgroups. These computations are possible since one understands in the cases of interest the geometry of \underline{E}G and G\\underline{E}G. We focus on the case described in Subsection 4.11, namely of a discrete group G satisfying the conditions (M) and (NM). Let s: EG → \underline{E}G be the up to G-homotopy unique G-map. Denote by j_{i}: M_{i} → G the inclusion.

8.1 Group Homology

We begin with the group homology H_{n}(BG) (with integer coefficients). Let H̃_{p}(X) be the reduced homology, i.e. the kernel of the map H_{n}(X) → H_{n}({pt.}) induced by the projection X → {pt.}. The Mayer-Vietoris sequence applied to the pushout, which is obtained from the G-pushout (4.17) by dividing out the G-action, yields the long Mayer-Vietoris sequence

… → H_{p+1}(G\\underline{E}G)) ⟶^{∂_{p+1}} ⨁_{i∈I} H̃_{p}(BM_{i}) \underrightarrow{⨁_{i∈I} H_{p}(Bj_{i})} H_{p}(BG)

⟶^{H_{p}(G\s)} H_{p}(G\\underline{E}G) ⟶^{∂_{p}} … (8.1)

In particular we obtain an isomorphism for p ≥ dim(\underline{E}G) + 2

⨁_{i∈I} H_{p}(Bj_{i}): ⨁_{i∈I} H̃_{p}(BM_{i}) ⟶^{≅} H_{p}(BG).  (8.2)

This example and the forthcoming ones show why it is important to get upper bounds on the dimension of \underline{E}G and to understand the quotient space G\\underline{E}G. For Fuchsian groups and for one-relator groups we have dim(G\\underline{E}G) ≤ 2 and it is easy to compute the homology of G\\underline{E}G in this case as explained in Section 6.

8.2 Topological K-Theory of Group C^{∗}-Algebras

Analogously one can compute the source of the assembly map appearing in the Baum-Connes Conjecture 7.1. Namely, the Mayer-Vietoris sequence associated to the G-pushout (4.17) and the one associated to its quotient under the Gaction look like

… → K_{p+1}^{G}(\underline{E}G) → ⨁_{i∈I} K_{p}^{G}(G ×_{M_{i}} EM_{i})

→ (⨁_{i∈I} K_{p}^{G}(G/M_{i})) ⨁ K_{p}^{G}(EG) → K_{p}^{G}(\underline{E}G) → … (8.3)

and

… → K_{p+1}(G\\underline{E}G) → ⨁_{i∈I} K_{p}(BM_{i})

→ (⨁_{i∈I} K_{p}(\{pt.\})) ⨁ K_{p}(BG) → ⨁_{i∈I} K_{p}(G\\underline{E}G) → … (8.4)

Notice that for a free G-CW -complex X there is a canonical isomorphisms K_{p}^{G}(X) ≅ K_{p}(G\X). We can splice these sequences together and obtain the long exact sequence

… → K_{p+1}(G\\underline{E}G) → ⨁_{i∈I} K_{p}^{G}(G/M_{i}) → ⨁_{i∈I} K_{p}(\{pt.\}) ⨁ K_{p}^{G}(\underline{E}G)

→ K_{p}(G\\underline{E}G) → … (8.5)

There are identification of K_{0}^{G}(G/M_{i}) with the complex representation ring R_{ℂ}(M_{i}) of the finite group M_{i} and of K_{0}({pt.}) with ℤ. Under these identification the map K_{0}^{G}(G/M_{i}) → K_{0}({pt.}) becomes the split surjective map ϵ: R_{ℂ}(M_{i}) → ℤ which sends the class of a complex M_{i}-representation V to the complex dimension of ℂ⊗_{ℂ[M_{i}]} V. The kernel of this map is denoted by R̃_{ℂ}(M_{i}). The groups K_{1}^{G}(G/M_{i}) and K_{1}({pt.}) vanish. The abelian group R_{ℂ}(M_{i}) and hence also R̃_{ℂ}(M_{i}) are finitely generated free abelian groups. If ℤ ⊆ Λ ⊆ ℚ is ring such that the order of any finite subgroup of G is invertible in Λ, then the map

Λ ⊗_{ℤ} K_{p}^{G}(s): Λ ⊗_{ℤ} K_{p}^{G}(EG) → Λ ⊗_{ℤ} K_{p}(G\\underline{E}G)

is an isomorphism for all p ∈ ℤ [58, Lemma 2.8 (a)]. Hence the long exact sequence (8.5) splits after applying Λ ⊗_{ℤ} −. We conclude from the long exact sequence (8.5) since the representation ring of a finite group is torsionfree

Theorem 8.6. Let G be a discrete group which satisfies the conditions (M) and (NM) appearing in Subsection 4.11. Suppose that the Baum-Connes Conjecture 7.1 is true for G. Let {(M_{i}) | i ∈ I} be the set of conjugacy classes of maximal finite subgroups of G. Then there is an isomorphism

K_{1}(C_{r}^{∗}(G)) ⟶^{≅} K_{1}(G\\underline{E}G)

and a short exact sequence

0 → ⨁_{i∈I} R̃_{ℂ}(M_{i}) → K_{0}(C_{r}^{∗}(G)) → K_{0}(G\\underline{E}G) → 0,

which splits if we invert the orders of all finite subgroups of G.

8.3 Algebraic K-and L-Theory of Group Rings

Suppose that G satisfies the Farrell-Jones Conjecture 7.2. Then the computation of the relevant groups K_{n}(RG) or L_{n}^{⟨−∞⟩}(RG) respectively is equivalent to the computation of H_{n}^{G}(E_{VCYC}(G), K) or H_{n}^{G}(E_{VCYC}(G), L^{−∞}) respectively. The following result is due to Bartels [5]. Recall that \underline{E}G is the same as E_{FIN} (G).

Theorem 8.7. (i) For every group G, every ring R and every n ∈ ℤ the up to G-homotopy unique G-map f: E_{FIN} (G) → E_{VCYC}(G) induces a split injection

H_{n}^{G}(f; K_{R}): H_{n}^{G}(E_{FIN} (G); K_{R}) → H_{n}^{G}(E_{VCYC}(G); K_{R}); (ii) Suppose R is such that K_{−i}(RV) = 0 for all virtually cyclic subgroups V

of G and for sufficiently large i (for example R = ℤ will do). Then we get a split injection

H_{n}^{G}(f; L_{R}^{⟨−∞⟩}): H_{n}^{G}(E_{FIN} (G); L_{R}^{⟨−∞⟩}) → H_{n}^{G}(E_{VCYC}(G); L_{R}^{⟨−∞⟩}).

It remains to compute H_{n}^{G}(E_{FIN} (G); K) and H_{n}^{G}(E_{VCYC}(G), E_{FIN} (G); K), if we arrange f to be a G-cofibration and think of E_{FIN} (G) as a G-CW - subcomplex of E_{VCYC}(G). Namely, we get from the Farrell-Jones Conjecture 7.2 and Theorem 8.7 an isomorphism

H_{n}^{G}(E_{FIN} (G); K) ⨁ H_{n}^{G}(E_{VCYC}(G), E_{FIN} (G); K) ⟶^{≅} K_{n}(RG).

The analogous statement holds for L_{R}^{⟨−∞⟩}), provided R satisfies the conditions appearing in Theorem 8.7 (ii).

Analogously to Theorem 8.6 one obtains

Theorem 8.8. Let G be a discrete group which satisfies the conditions (M) and (NM) appearing in Subsection 4.11. Let {(M_{i}) | i ∈ I} be the set of conjugacy classes of maximal finite subgroups of G. Then

(i) There is a long exact sequence

… → H_{p+1}(G\E_{FIN} (G); K(R)) → ⨁_{i∈I} K_{p}(R[M_{i}])

→ ⨁_{i∈I} K_{p}(R) ⨁ H_{p}^{G}(E_{FIN} (G); K_{R}) → ⨁ H_{p}(G\E_{FIN} (G); K(R)) → …

and analogously for L_{R}^{⟨−∞⟩}.

(ii) For R = ℤ there are isomorphisms

⨁_{i∈I} Wh_{n}(M_{i}) ⨁ H_{n}^{G}(E_{FIN} (G), E_{VCYC}(G); K_{ℤ}) ⟶^{≅} Wh_{n}(G).

Remark 8.9. These results about groups satisfying conditions (M) and (NM) are extended in [53] to groups which map surjectively to groups satisfying conditions (M) and (NM) with special focus on the semi-direct product of the discrete three-dimensional Heisenberg group with ℤ/4.

Remark 8.10. In [70] a special model for \underline{E}SL_{3}(ℤ) is presented which allows to compute the integral group homology. Information about the algebraic Ktheory of SL_{3}(ℤ) can be found in [72, Chapter 7], [79].

The analysis of the other term H_{n}^{G}(E_{VCYC}(G), E_{FIN} (G); K) simplifies considerably under certain assumptions on G.

Theorem 8.11 (On the structure of E_{VCYC}(G)). Suppose that G satisfies the following conditions:

• Every infinite cyclic subgroup C ⊆ G has finite index in its centralizer C_{G}C;

• There is an upper bound on the orders of finite subgroups.

(Each word-hyperbolic group satisfies these two conditions.) Then

(i) For an infinite virtually cyclic subgroup V ⊆ G define

V_{max} = ⋃ {N_{G}C | C ⊂ V infinite cyclic normal}.

Then

(a) V_{max} is an infinite virtually cyclic subgroup of G and contains V;

(b) If V ⊆ W ⊆ G are infinite virtually cyclic subgroups of G, then V_{max} = W_{max};

(c) Each infinite virtually cyclic subgroup V is contained in a unique maximal infinite virtually cyclic subgroup, namely V_{max}, and N_{G}V_{max} = V_{max};

(ii) Let {V_{i} | i ∈ I} be a complete system of representatives of conjugacy classes of maximal infinite virtually cyclic subgroups. Then there exists a G-pushout

∐_{i∈I} G ×_{V_{i}} E_{FIN} (V_{i}) ⟶ E_{FIN} (G)
pr↓  ↓
∐_{i∈I} G/V_{i}  ⟶ E_{VCYC}(G)

whose upper horizontal arrow is an inclusion of G-CW -complexes.

(iii) There are natural isomorphisms

⨁_{i∈I} H_{n}^{V_{i}}(E_{VCYC}(V_{i}), E_{FIN} (V_{i}); K_{R}) ⟶^{≅} H_{n}^{G}(E_{VCYC}(G), E_{FIN} (G); K_{R})
⨁_{i∈I} H_{n}^{V_{i}}(E_{VCYC}(V_{i}), E_{FIN} (V_{i}); L_{R}^{⟨−∞⟩}) ⟶^{≅} H_{n}^{G}(E_{VCYC}(G), E_{FIN} (G); L_{R}^{⟨−∞⟩}).

Proof. Each word-hyperbolic group G satisfies these two conditions by [13, Theorem 3.2 in III.Γ.3 on page 459 and Corollary 3.10 in III.Γ.3 on page 462].

(i) Let V be an infinite virtually cyclic subgroup V ⊆ G. Fix a normal infinite cyclic subgroup C ⊆ V. Let b be a common multiple of the orders of finite subgroups of G. Put d := b · b!. Let e be the index of the infinite cyclic group dC = {d · x | x ∈ C} in its centralizer C_{G}dC. Let D ⊂ dC be any non-trivial subgroup. Obviously dC ⊆ C_{G}D. We want to show

[C_{G}D : dC] ≤ b · e^{2}.  (8.12)

Since D is central in C_{G}D and C_{G}D is virtually cyclic and hence |C_{G}D/D| < ∞, the spectral sequence associated to the extension 1 → D → C_{G}D → C_{G}D/D → 1 implies that the map D = H_{1}(D) → H_{1}(C_{G}D) is injective and has finite cokernel. In particular the quotient of H_{1}(C_{G}D) by its torsion subgroup H_{1}(C_{G}D)/ tors is an infinite cyclic group. Let p_{C_{G}D}: C_{G}D → H_{1}(C_{G}D)/ tors be the canonical epimorphism. Its kernel is a finite normal subgroup. The following diagram commutes and has exact rows

1 ⟶ ker(p_{C}) ⟶ C_{G}C ⟶^{p_{C}} H_{1}(C_{G}C)/ tors ⟶ 1
↓  ↓  ↓
1 ⟶ ker(p_{D}) ⟶ C_{G}D ⟶^{p_{D}} H_{1}(C_{G}D)/ tors ⟶ 1

where the vertical maps are induced by the inclusions C_{G}C ⊆ C_{G}D. All vertical maps are injections with finite cokernel. Fix elements z_{C} ∈ C_{G}C and z_{D} ∈ C_{G}D such that p_{C}(z_{C}) and p_{D}(z_{D}) are generators. Choose l ∈ ℤ such that p_{C}(z_{C}) is send to l · p_{D}(z_{D}). Then there is k ∈ ker(p_{D}) with z_{C} = k · z_{D}^{l}. The order of ker(p_{D}) divides b by assumption. If ϕ: ker(p_{D}) → ker(p_{D}) is any automorphism, then ϕ^{b!} = id. This implies for any element k ∈ ker(p_{D}) that

∏_{i=0}^{d−1} ϕ^{i}(k) = ( ∏_{i=0}^{b!−1} ϕ^{i}(k))^{b} = 1.

Hence we get in C_{G}D if ϕ is conjugation with z_{D}^{l}

z_{C}^{d} = (k · z_{D}^{l})^{d} = ∏_{i=0}^{d−1} ϕ^{i}(k) · z_{D}^{dl} = z_{D}^{dl}.

Obviously z_{D} ∈ C_{G}dC since z_{C}^{d} = z_{D}^{dl} generates dC. Hence z_{D}^{e} lies in dC and we get z_{D}^{e} = z_{C}^{df} for some integer f. This implies z_{D}^{e} = z_{D}^{ldf} and hence that l divides e. We conclude that the cokernel of the map H_{1}(C_{G}C)/ tors → H_{1}(C_{G}D)/ tors is bounded by e. Hence the index [C_{G}D : C_{G}C] is bounded by b·e since the order of ker(p_{D}) divides b. Since dC ⊆ C_{G}C ⊆ C_{G}dC ⊆ C_{G}D holds, equation 8.12 follows.

Next we show that there is a normal infinite cyclic subgroup C_{0} ⊆ V such that V_{max} = N_{G}C_{0} holds. If C′ and C′′ are infinite cyclic normal subgroups of V, then both C_{G}C′ and C_{G}C′′ are contained in C_{G}(C′ ∩ C′′) and C′ ∩ C′′ is again an infinite cyclic normal subgroup. Hence there is a sequence of normal infinite cyclic subgroups of V

dC ⊇ C_{1} ⊇ C_{2} ⊇ C_{3} ⊇ …

which yields a sequence C_{G}dC ⊆ C_{G}C_{1} ⊆ C_{G}C_{2} ⊂ … satisfying

⋃ {C_{G}C_{n} | n ≥ 1} = ⋃ {C_{G}C | C ⊂ V infinite cyclic normal}.

Because of 8.12 there is an upper bound on [C_{G}C_{n} : C_{G}dC] which is independent of n. Hence there is an index n_{0} with

C_{G}C_{n_{0}} = ⋃ {C_{G}C | C ⊂ V infinite cyclic normal}.

For any infinite cyclic subgroup C ⊆ G the index of C_{G}C in N_{G}C is 1 or 2. Hence there is an index n_{1} with

N_{G}C_{n_{1}} = ⋃ {N_{G}C | C ⊂ V infinite cyclic normal}.

Thus we have shown the existence of a normal infinite cyclic subgroup C ⊆ V with V_{max} = C. Now assertion (i)a follows.

We conclude assertion (i)b from the fact that for an inclusion of infinite virtually cyclic group V ⊆ W there exists a normal infinite cyclic subgroup C ⊆ W such that C ⊆ V holds. Assertion (i)c is now obviously true. This finishes the proof of assertion (i).

(ii) Construct a G-pushout

∐_{i∈I} G ×_{V_{i}} E_{FIN} (V_{i}) ⟶^{j} E_{FIN} (G)
pr↓  ↓
∐_{i∈I} G/V_{i}  ⟶  X

with j an inclusion of G-CW -complexes. Obviously X is a G-CW -complex whose isotropy groups are virtually cyclic. It remains to prove for virtually cyclic H ⊆ G that X^{H} is contractible.

Given a V_{i}-space Y and a subgroup H ⊆ G, there is after a choice of a map of sets s: G/V_{i} → G, whose composition with the projection G → G/V_{i} is the identity, a G-homeomorphism

∐_{\begin{matrix}w∈G/V_{i} \\ s(w)^{−1}Hs(w)⊆V_{i} \end{matrix}} Y^{s(w)^{−1}Hs(w)} ⟶^{≅} (G ×_{V_{i}} Y)^{H} ,  (8.13)

which sends y ∈ Y^{s(w)^{−1}Hs(w)} to (s(w), y).

If H is infinite, the H-fixed point set of the upper right and upper left corner is empty and of the lower left corner is the one-point space because of assertion (i)c and equation 8.13. Hence X^{H} is a point for an infinite virtually cyclic subgroup H ⊆ G.

If H is finite, one checks using equation 8.13 that the left vertical map induces a homotopy equivalence on the H-fixed point set. Since the upper horizontal arrow induces a cofibration on the H-fixed point set, the right vertical arrow induces a homotopy equivalence on the H-fixed point sets. Hence X^{H} is contractible for finite H ⊆ G. This shows that X is a model for E_{VCYC}(G).

(iii) follows from excision and the induction structure. This finishes the proof of Theorem 8.11.

Theorem 8.11 has also been proved by Daniel Juan-Pineda and Ian Leary [39] under the stronger condition that every infinite subgroup of G, which is not virtually cyclic, contains a non-abelian free subgroup. The case, where G is the fundamental group of a closed Riemannian manifold with negative sectional curvature is treated in [6].

Remark 8.14. In Theorem 8.11 the terms H_{n}^{V_{i}}(E_{VCYC}(V_{i}), E_{FIN} (V_{i}); K_{R}) and H_{n}^{V_{i}}(E_{VCYC}(V_{i}), E_{FIN} (V_{i}); L_{R}^{⟨−∞⟩}) occur. They also appear in the direct sum decomposition

K_{n}(RV_{i}) ≅ H_{n}^{V_{i}}(E_{FIN} (V_{i}); K_{R}) ⨁ H_{n}^{V_{i}}(E_{VCYC}(V_{i}), E_{FIN} (V_{i}); K_{R});
L_{n}(RV_{i}) ≅ H_{n}^{V_{i}}(E_{FIN} (V_{i}); L_{R}^{⟨−∞⟩}) ⨁ H_{n}^{V_{i}}(E_{VCYC}(V_{i}), E_{FIN} (V_{i}); L_{R}^{⟨−∞⟩}).

They can be analysed further and contain information about and are build from the Nil and UNIL-terms in algebraic K-theory and L-theory of the infinite virtually cyclic group V_{i}. They vanish for L-theory after inverting 2 by results of [19]. For R = ℤ they vanishes rationally for algebraic K-theory by results of [45].

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Notation

cd(G), 30 cd(M), 30 CX, 34 C_{r}^{∗}(G), 42 E_{F}(G), 7 \underline{E}G, 7 F_{n}, 25 FP_{n}, 30 FP_{∞}, 30 G^{0}, 14 G_{d}, 32 \overline{G}, 14 J_{F}(G), 8 \underline{J}G, 8 l(H), 34 N_{G}H, 34 Or(G), 31 Or_{F}(G), 31 Out(F_{n}), 25 P C_{0}(G), 18 P_{d}(G, S), 25 T = T (G, X, X_{0}), 22 vcd(G), 37 W_{G}H, 34 [X, Y]^{G}, 5 VF, 40 \underline{ℤ}, 32 π = π(G, X, X_{0}), 21 Γ_{g,r}^{s}, 26 ALL, 7 COM, 7 COMOP, 7 FIN, 7 T_{g,r}^{s}, 26 TR, 7 VCYC, 7

Index

apartments, 23

building affine, 23 Euclidean, 23

cell

equivariant closed n-dimensional

cell, 4 equivariant open n-dimensional

cell, 4 cellular

map, 7 classifying G-CW -complex for a fam-

ily of subgroups, 7 classifying numerable G-space for a

family of subgroups, 8 cocompact, 5 cohomological dimension

for groups, 30 for modules, 30

colimit topology, 4 component group, 14 component of the identity, 14 Conjecture

Baum-Connes Conjecture, 42 Farrell-Jones Conjecture, 43

δ-hyperbolic group, 25 metric space, 24

equivariant smooth triangulation, 5

F-numerable G-space, 8 family of subgroups, 7 fundamental group of a graph of groups,

21

G-CW -complex, 4 finite, 5 finite dimensional, 5 of dimension ≤ n, 5 of finite type, 5 graph of groups, 21

fundamental group, 21 group

almost connected, 14

hyperbolic, 25 poly-ℤ, 38 totally disconnected, 13 virtually poly-cyclic, 38

Hirsch rank, 38 hyperbolic

group, 25

metric space, 25

length of a subgroups, 34

mapping class group, 26 marked metric graph, 26 metric space

hyperbolic, 25

numerable G-principal bundle, 9 universal, 9

orbit category, 31 outer space, 25

spine of, 26

proper G-space, 4

resolution finite, 30 free, 30 resolution, 30

Rips complex, 25

skeleton, 4

Teichmüller space, 26

Theorem

A criterion for 1-dimensional mod-

els, 36 Actions on CAT(0)-spaces, 19 Actions on simply connected non-

positively curved manifolds, 19

Actions on trees, 20 Affine buildings, 24 Algebraic and geometric finite-

ness conditions, 32 Almost connected groups, 19 Comparison of E_{F}(G) and J_{F}(G),

14 Completion Theorem for discrete

groups, 44 Discrete subgroups of almost con-

nected Lie groups, 19 Discrete subgroups of Lie groups,

38 Equivariant Cellular Approxima-

tion Theorem, 7 Finiteness conditions for BG,

30 Homotopy characterization of E_{F}(G),

7 Homotopy characterization of J_{F}(G),

8 Models based on actions on trees,

21 On the structure of E_{VCYC}(G),

48 Operator theoretic model, 18 Passage from topological groups

to discrete groups, 33 Passage from topological groups

to totally disconnected groups, 33

Passage from totally disconnected groups to discrete groups, 33

Rips complex, 25 The homotopy type of G\\underline{E}G,

41 Virtual cohomological dimension

and dim(\underline{E}G, 37 Virtual cohomological dimension

and the condition B(d), 37 Whitehead Theorem for Fami-

lies, 5 tree, 20 type F P_{n}

for groups, 30 for modules, 30

type F P_{∞} for groups, 30 for modules, 30

type VF, 40

universal G-CW -complex for proper G-actions, 7

universal G-space for proper G-actions,

8 universal numerable G-space for proper

G-actions, 8

virtual cohomological dimension, 37

weak homotopy equivalence, 6

weakly contractible, 6

ℤOr_{F}(G)-module, 31 free, 31