Survey on Classifying Spaces for Families of Subgroups Wolfgang Lück* Fachbereich Mathematik Universität Münster Einsteinstr. 62 48149 Münster Germany December 19, 2003 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 EF (G) and the numerable G-space version JF (G). They agree if G is discrete, or if G is a Lie group and each element in F compact, or if G is totally disconnected and F is the family of compact subgroups or of compact open 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 K- and 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 EF (G) and the numerable G-space version JF (G). They agree, if G is discrete, or if G is a Lie group_and_each_element_in_F_compact, or if G is totally disconnected and F *email: lueck@math.uni-muenster.de www: http://www.math.uni-muenster.de/u/lueck/ FAX: 49 251 8338370 1 is the family of compact subgroups or of compact open subgroups, but not in general 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 EF (G) and JF (G) and in particular for the orbit space G\EFIN (G). The space EF (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, one- relator 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\EFIN (G) are homotopy equivalent. We deal with the question whether there are finite models, models of fi- nite type, finite dimensional models. In some sense the algebra of a discrete group G is reflected in the geometry of the spaces EFIN (G). For torsionfree discrete groups EFIN (G) is the same as EG. For discrete groups with torsion the space EFIN (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 EFIN (G) and the minimal dimension is related to the notion of virtual cohomological dimension. The space JCOM (G) associated to the family of compact subgroups is some- times also called the classifying space for proper group actions. We will abbre- viate it as J_G. For a discrete group G it agrees with EFIN (G) which we will abbreviate by E_G. Sometimes the abbreviation E_G is used in the literature, especially in connection with the Baum-Connes Conjecture, also for a topologi- cal group, but corresponds in the notion used in this article to J_G = JCOM (G). There is no difference between E_G and J_G, if G is discrete, a Lie group, or totally disconnected. 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 2 useful comments. Contents 0 Introduction 1 1 G-CW -Complex-Version 5 1.1 Basics about G-CW -Complexes . . . . . . . . . . . . . . . . . . . 5 1.2 The G-CW -Version for the Classifying Space for a Family . . . . 7 2 Numerable G-Space-Version 9 3 Comparison of the Two Versions 12 4 Special Models 17 4.1 Operator Theoretic Model . . . . . . . . . . . . . . . . . . . . . . 17 4.2 Almost Connected Groups . . . . . . . . . . . . . . . . . . . . . . 17 4.3 Actions on Simply Connected Non-Positively Curved Manifolds . 18 4.4 Actions on CAT(0)-spaces . . . . . . . . . . . . . . . . . . . . . . 18 4.5 Actions on Trees and Graphs of Groups . . . . . . . . . . . . . . 18 4.6 Affine Buildings . . . . . . . . . . . . . . . . . . . . . . . . . . . * *. 22 4.7 The Rips Complex of a Word-Hyperbolic Group . . . . . . . . . 23 4.8 Arithmetic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . * *24 4.9 Outer Automorphism Groups of Free groups . . . . . . . . . . . . 24 4.10 Mapping Class groups . . . . . . . . . . . . . . . . . . . . . . . . 25 4.11 Groups with Appropriate Maximal Finite Subgroups . . . . . . . 25 4.12 One-Relator Groups . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.13 Special Linear Groups of (2,2)-Matrices . . . . . . . . . . . . . . 27 4.14 Manifold Models . . . . . . . . . . . . . . . . . . . . . . . . . . . * *28 5 Finiteness Conditions 29 5.1 Review of Finiteness Conditions on BG . . . . . . . . . . . . . . 29 5.2 Modules over the Orbit Category . . . . . . . . . . . . . . . . . . 30 5.3 Reduction from Topological Groups to Discrete Groups . . . . . 32 5.4 Poset of Finite Subgroups . . . . . . . . . . . . . . . . . . . . . . * *33 5.5 Extensions of Groups . . . . . . . . . . . . . . . . . . . . . . . . . * *35 5.6 One-Dimensional Models for E_G . . . . . . . . . . . . . . . . . . 36 5.7 Groups of Finite Virtual Dimension . . . . . . . . . . . . . . . . 36 5.8 Counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . * *39 6 The Orbit Space of E_G 41 3 7 Relevance and Applications of Classifying Spaces for Families 43 7.1 Baum-Connes Conjecture . . . . . . . . . . . . . . . . . . . . . . 43 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 . . . . . . . . . . . . . . 45 8 Computations using Classifying Spaces for Families 46 8.1 Group Homology . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 8.2 Topological K-Theory of Group C*-Algebras . . . . . . . . . . . 46 8.3 Algebraic K-and L-Theory of Group Rings . . . . . . . . . . . . 47 References 52 Notation 58 Index 59 4 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 X0 X1 . . .Xn . . . Xn = X n 0 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 \ Xn is closed in Xn for all n 0) and Xn is obtained from Xn-1 for each n 0 by attaching equivariant n-dimensional cells, i.e. there exists a G-pushout ` n-1 ` i2Inqni i2InG=Hix?S ------! Xn-1? ?y ?y ` n i2InG=Hix D ------!`i2InXn Qi n The space Xn 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 Xn - Xn-1, i.e. the preimage of a path component of G\(Xn - Xn-1). The closure of an equivariant open n-dimensional cell is called an equivari- ant closed n-dimensional cell. If one has chosen the G-pushouts in Defini- tion 1.1, then the equivariant open n-dimensional cells are the G-subspaces Qi(G=Hix (Dn - Sn-1)) and the equivariant closed n-dimensional cells are the G-subspaces Qi(G=Hix Dn). 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 Vx of x and Wy of y in X such that the closure of the subset {g 2 G | gVx \ Wy 6= ;} of G is compact. A G-CW -complex X is proper if and only if all its isotropy groups are compact [47, 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 X0 X1 . . .Xn . . . Xn = X. n 0 5 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 2 G with ge \ e 6= ; 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 2 G with ge \ e 6= ; 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 X0 by simplicial automorphisms. The filtration of the barycentric subdivision X0 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 n- skeleton is finite. It is called of dimension n if X = Xn and finite dimensio* *nal 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] 7! [g O f] between the set of G-homotopy classes of G-maps is bijective; (ii)For any H 2 F the map fH :Y H ! ZH is a weak homotopy equivalence i.e. the map ßn(fH , y): ßn(Y H, y) ! ßn(ZH , fH (y)) is bijective for any base point y 2 Y H and n 2 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 x A, Y ]G -! [A, Y H]. Hence for any CW -complex A the map fH induces a bijection (fH )*: [A, Y H] ! [A, ZH ], [g] ! [g O fH ]. 6 This is equivalent to fH being a weak homotopy equivalence by the classical non-equivariant Whitehead Theorem [79, 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 [73, Proposition II.2.6 on page 107]. We have to show for any G-CW -complex X such that two G-maps f0, f1: 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. ßn(Y H, y) consists of one element for all * *base points y 2 Y H. Since X is colimn!1 Xn it suffices to construct inductively over n G-homotopies h[n]: Xn x[0, 1] ! Z such that h[n]i= fiholds for i = 0, 1 and h[n]|Xn-1x[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 ` ` n-1 i2Inqni i2InG=Hix?S ------! Xn-1? ?y ?y ` n i2InG=Hix D -`-----!i2InXn Qi n One easily checks that the desired G-homotopy h[n] exists if and only if we can find for each i 2 I an extension of the G-map f0 O Qni[ f1 O Qni[ h[n - 1] O (qnix id[0,1]): G=Hix Dn x {0} [ G=Hix Dn x {1} [ G=Hix Sn-1 x [0, 1] ! Y to a G-map G=Hix Dn x [0, 1] ! Y . This is the same problem as extending the (non-equivariant) map Dn x {0} [ Dn x {1} [ Sn-1 x [0, 1] ! Y , which is given by restricting the G-map above to 1Hi, to a (non-equivariant) map Dn x [0, 1] ! Y Hi. Such an extension exists since Y Hiis 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 fH :XH ! 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(Xn) Yn 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 [73, Theorem II.2.1 on page 104]. 1.2 The G-CW -Version for the Classifying Space for a Family 7 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 EF (G) for the classifying G- CW -complex for the family F of subgroups is a G-CW -complex EF (G) which has the following properties: i.) All isotropy groups of EF (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 E_G := ECOM (G) and call it the universal G-CW -complex for proper G-actions. In other words, EF (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 EF (G) are G-homotopy equivalent and for two families F0 F1 there is up to G-homotopy precisely one G-map EF0(G) ! EF1(G). Theorem 1.9 (Homotopy characterization of EF (G)). Let F be a family of subgroups. (i)There exists a model for EF (G) for any family F; (ii)A G-CW -complex X is a model for EF (G) if and only if all its isotropy groups belong to F and for each H 2 F the H-fixed point set XH is weakly contractible. Proof.(i) A model can be obtained by attaching equivariant cells G=H xDn for all H 2 F to make the H-fixed point sets weakly contractible. See for instance [47, 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 EALL(G) is G=G. In Section 4 we will give many interesting geometric models for classifying spaces EF (G), in particular for the case, whe* *re G is discrete and F = FIN or, more generally, where G is a (locally compact topological Hausdorff) group and F = COM. In some sense E_G = ECOM (G) is the most interesting case. 8 2 Numerable G-Space-Version In this section we explain the numerable G-space version of the classifying spa* *ce for a family F of subgroups of group G. Definition 2.1 (F-numerable G-space). A F-numerable G-space is a G- space, for which there exists an open covering {Ui| i 2 I} by G-subspaces such that there is for each i 2 I a G-map Ui ! G=Gi for some Gi 2 F and there is a locally finite partition of unity {ei | i 2 I} subordinate to {Ui | i 2 I}* * by G-invariant functions ei: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 all its isotropy groups belong to F. Proof.This follows from the Slice Theorem for G-CW -complexes [47, Theorem __ 1.37] and the fact that G\X is a CW -complex and hence paracompact [60]. |__| Definition 2.3 (Classifying numerable G-space for a family of sub- groups). Let F be a family of subgroups of G. A model JF (G) for the classi- fying numerable G-space for the family F of subgroups is a G-space which has the following properties: i.) JF (G) is F-numerable. ii.) For any F-numerable G-space X there is up to G-homotopy precisely one G-map X ! JF (G). We abbreviate J_G := JCOM (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, JF (G) is a terminal object in the G-homotopy category of F-numerable G-spaces. In particular two models for JF (G) are G-homotopy equivalent, and for two families F0 F1 there is up to G-homotopy precisely one G-map JF0(G) ! JF1(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 JF (G)). Let F be a family of subgroups. (i)There exists a model for JF (G) for any family F; (ii)Let X be a F-numerable G-space. Equip X x X with the diagonal action and let pri:X x X ! X be the projection onto the i-th factor for i = 1, 2. Then X is a model for JF (G) if and only if for each H 2 F there is x 2 X with H Gx and pr1and pr2are G-homotopic. (iii)For H 2 F the H-fixed point set JF (G)H is contractible. 9 Proof.(i) A model for JF (G) is constructed by tom Dieck [73,`Theorem I.6.6. on page 47], namely, as the infinite join *1n=1L for L = H2F G=H. There G is assumed to be compact but the proof goes through for locally compact topological Hausdorff groups. (ii) Let X be a model for the classifying space JF (G) for F. Then X x X with the diagonal G-action is a F-numerable G-space. Hence pr1and pr2are G-homotopic by the universal property. Since for any H 2 F the G-space G=H is F-numerable, there must exist a G-map G=H ! X by the universal property of JF (G). If x is the image under this map of 1H, then H Gx. Suppose that X is a G-space such that for each H 2 F there is x 2 X with H Gx and pr1and pr2are G-homotopic. We want to show that then X is a model for JF (G). Let f0, f1: Y ! X be two G-maps. Since priO(f0 x f1) = fi holds for i = 0, 1, f0 and f1 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 JF (G) it suffices to do this in the case, where Y = *1n=* *1L for L = H2F 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 pr1to pr2to construct a G-map *1n=1L ! X. (iii) Restricting to 1H yields a bijection ~= H H [G=H x JF (G)H , JF (G)]G -! [JF (G) , JF (G) ], where we consider XH as a G-space with trivial G action. Since G=H x XH is a F-numerable G-space, [JF (G)H , JF (G)H ] consists of one element.__Hence JF (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 JF (G) if XH is contractible for each H 2 F. Example 2.7 (Numerable G-principal bundles). A numerable (locally triv- ial) G-principal bundle p: E ! B consists by definition of a TR-numerable G- space 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] [73, Chapter I Section 8]. If p: E ! B is a universal numerable G-principal bundle, then E is a model for JTR (G). Conversely, JTR (G) ! G\JTR (G) is a model for the universal numerable G-principal bundle. We conclude that a TR-numerable G-space X is a model for JTR (G) if and only if X is contractible (compare Remark 2.6). 10 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 [60], it is automatically a numerable G-principal bundle. The CW -complex structure on B pulls back to G-CW -structure on E [47, 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 JTR (G) ! G\JTR (G) lifts to a G-bundle map from p to ETR (G) ! G\ETR (G) and two such G-bundle maps from p to ETR (G) ! G\ETR (G) are G-bundle homotopic. Hence for G-principal bundles over CW -complexes one can use ETR (G) ! G\ETR (G) as the universal object. We will compare the spaces EF (G) and JF (G) in Section 3. In Section 4 we will give many interesting geometric models for EF (G) and JF (G) in particular in the case F = COM. In some sense J_G = JCOM (G) is the most interesting case. 11 3 Comparison of the Two Versions In this section we compare the two classifying spaces EF (G) and JF (G). Since EF (G) is a F-numerable space by Lemma 2.2, there is up to G- homotopy precisely one G-map u: EF (G) ! JF (G). (3.1) Lemma 3.2. The following assertions are equivalent for a family F of subgroups of G: (i)The map u: EF (G) ! JF (G) defined in 3.1 is a G-homotopy equivalence; (ii)The G-spaces EF (G) and JF (G) are G-homotopy equivalent; (iii)The G-space JF (G) is G-homotopy equivalent to a G-CW -complex, whose isotropy groups belong to F; (iv)There exists a G-map JF (G) ! Y to a G-CW -complex Y , whose isotropy groups belong to F; Proof.This follows from the universal properties of EF (G) and JF (G). |___| Lemma 3.3. Suppose either that every element H 2 F is an open (and closed) subgroup of G or that G is a Lie group and F COM. Then the map u: EF (G) ! JF (G) defined in 3.1 is a G-homotopy equivalence. Proof.We have to inspect the construction in [73, Lemma 6.13 in Chapter II on page 49] and will use the`same notation as in that paper. Let Z be a F- numerable G-space. Let X = H2F G=H. Then *1n=1X is a model for JF (G) by [73, Lemma 6.6 in Chapter II on page 47]. We inspect the construction of a G-map f :Z ! *1n=1X. One constructs a countable covering {Un | n = 1, 2, . .}. of Z by G-invariant open subsets of Z together with a locally finite subordinate partition of unity {vn | n = 1, 2, . .}.by G-invariant functions vn :Z ! [0, 1] and G-maps OEn :Un ! X. Then one obtains a G-map f :Z ! *1n=1X, z 7! (v1(z)OE1(z), v2(z)OE2(z), . .)., where vn(z)OEn(z) means 0x for any x 2 X if z 62 Un. Let ik: *kn=1X ! *1n=1X and jk: *kn=1X ! *k+1n=1X be the obvious inclusions. Denote by ffk: *kn=1X ! colimk!1 *kn=1X the structure map and by i: colimk!1 *kn=1X ! *1n=1X the map induced by the system {ik | k = 1, 2, . .}.. This G-map is a (continuous) bijective G-map but not necessarily a G-homeomorphism. Since the partition {vn | n = 1, 2, . .}.is locally finite, we can find for each z 2 Z an open G- invariant neighborhood Wz of z in Z and a positive integer kz such that vn vanishes on Wz for n > kz. Define a map f0z:Wz ! *kzn=1X, z 7! (v1(z)OE1(z), v2(z)OE2(z), . .,.vkz(z)OEkz(z)). 12 Then ffkzOf0z:Wz ! colimk!1 *kn=1X is a well-defined G-map whose composi- tion with i: colimk!1 *kn=1X ! *1n=1X is f|Wz . Hence the system of the maps ffkzO f0zdefines a G-map f0:Z ! colimk!1 *kn=1X such that i O f0 = f holds. Let Xn Yk n-1 = {(t1, t2. .t.n) | ti2 [0, 1], ti= 1} [0, 1] i=1 n=1 be the standard (n - 1)-simplex. Let _ k ! Y p: X x n ! *kn=1X, (x1, . .,.xn), (t1, . .,.tn) 7! (t1x1, . .,.tnxn) n=1 be the obvious projection. It is a surjective continuous map but in general not an identification. Let _*kn=1X be the topological space whose underlying set is the same as for *kn=1X but whose topology is the quotient topology with respect to p. The identity induces a (continuous) map _*kn=1X ! *kn=1X which is not a homeomorphism in general. Choose for n 1 a (continuous) function OEn :[0, 1] ! [0, 1] which satisfies OE-1n(0) = [0, 4-n ]. Define uk: *kn=1X ! _*kn=1X, _ fifi ! (tnxn | n = 1, . .,.k) 7! ___OEn(tn)__Pkxnfifin = 1,.. .,.k n=1 OEn(tn) fi _ _k _k+1 It is not hard to check that this G-map is continuous._If jk:*n=1X ! *n=1X is the obvious inclusion, we have uk+1 O jk = jkO uk for all k 1. Hence the system of the maps uk induces a G-map u: colimk!1 *kn=1X ! colimk!1 _*kn=1X. Next we want to show that each G-space _*kn=1X has the G-homotopy type of a G-CW -complex,iwhosejisotropy groups belong to F. We first show that _*k Q k n=1X is a n=1 G -CW -complex. It suffices to treat the case k = 2, the general case follows by induction over k. We can rewrite X_*X as a G x G- pushout X x X ---i1-!CX x X ? ? i2?y ?y X x CX ----! X_*X where CX is the cone over X and i1 and i2 are the obvious inclusions. Recall that we are working in the category of compactly generated spaces. Hence the 13 product of two G-CW -complexes is in a canonical way a (G x 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_*X inherits a (G x G)-CW -complex structure. i Q k j _k The problem is now to decide whether the n=1G -CW -complex *n=1X regarded as a G-space by the diagonal action has the G-homotopy type of a G- CW -complex. If each H 2 F is open, then each isotropy group of the G-space *kn=1X is open and we conclude from Remark 1.3 that _*kn=1X with the diagonal G-action is a G-CW -complex Suppose that G is a Lie group and eachQH 2 F is compact.iLemma 1.4 implies that for any compact subgroup K kn=1G Q k j the space n=1 G =K regarded as G-space by the diagonal action has the G-homotopy type of a G-CW -complex. We conclude from [47, Lemma 7.4 on page 121] that _*kn=1X with the diagonal G-action has the G-homotopy type of a G-CW -complex. The isotropy groups _*kn=1X belong to F since F is closed under_finite intersections and conjugation. It is not hard to check that each G- map jk is a G-cofibration. Hence colimk!1 _*kn=1X 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 G- map 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 topolog- ical Hausdorff) group G is called totally disconnected if it satisfies one of t* *he following equivalent conditions: (T) G is totally disconnected as a topological space, i.e. each component con- sists 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 neighbor- hoods. We have to explain why these three conditions are equivalent. The implica- tion (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 ECOMOP (G) ---u-! JCOMOP (G) ?? ? y ?y EF (G) ----!u JF (G) 14 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 h1, h2, . .,.hs in H such that H [si=1hiK. Put L := \h2H hKh-1. Then hLh-1 = L for all h 2 H. Since L = \si=1hiKh-1iholds, L is compact open. Hence LH is a compact open subgroup containing H. This implies that JF (G) is COMOP-numerable. Obviously JCOMOP (G) is F-numerable. We conclude from the universal properties that JCOMOP (G) ! JF (G) is a G-homotopy equivalence. The map u: ECOMOP (G) ! JCOMOP (G) is a G-homotopy equivalence by Lemma 3.3. This and Theorem 2.5 (iii) imply that ECOMOP (G)H is contractible for all H 2 F. Hence ECOMOP (G) ! EF (G) is a G-homotopy equivalence by Theorem_ 1.9 (ii). |__| Definition 3.6 (Almost connected group). Given a group G, let_G0 be the normal subgroup given by the component of the identity and G = G=G0 be_the component group. We call G almost connected if its component group 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 EF (G) and JF (G)). The map u: EF (G) ! JF (G) defined in 3.1 is a G-homotopy equivalence if one of the following con- ditions 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 2 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 (iii) and Theorem_ 4.3. |__| The following example shows that the map u: EF (G) ! JF (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: ETR (G) ! JTR (G) defined in 3.1 is a G- homotopy equivalence if and only if G is discrete. In view of Theorem 2.5 (iii) 15 and Lemma 3.3 this is equivalent to the statement that ETR (G) is contractible if and only if G is discrete. If G is discrete, we already know that ETR (G) is contractible. Suppose now that ETR (G) is contractible. We obtain a numerable G-principal bundle G ! ETR (G) ! G\ETR (G) by Remark 2.8. This implies that it is a fibration by a result of Hurewicz [79, Theorem on p. 33]. Since ETR (G) is contractible, G and the loop space (G\ETR (G)) are homotopy equivalent [79, 6.9* on p. 137, 6.10* on p. 138, Corollary 7.27 on p. 40]. Since G\ETR (G) is a CW -complex, (G\ETR (G)) has the homotopy type of a CW - complex [58]. Hence there exists a homotopy equivalence f :G ! X be from G to a CW -complex X. Then the induced map ß0(G) ! ß0(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. 16 4 Special Models In this section we present some interesting geometric models for the space EF (G) and JF (G) focussing on E_G and 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 EALL(G) and JALL(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 C0(G) be the Banach space of complex valued functions of G vanishing at infinity with the supremum-norm. The group G acts isometrically on C0(G) by (g . f)(x) := f(g-1x) for f 2 C0(G) and g, x 2 G. Let P C0(G) be the subspace of C0(G) consisting of functions f such that f is not identically zero and has non-negat* *ive 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 C0(G) is a model for J_G. Remark 4.2. Let G be discrete. Another model for J_G is the space X XG = {f :G ! [0, 1] | f has finite support, f(g) = 1} g2G with the topology coming from the supremum norm [7, page 248]. Let P1 (G) be the geometric realization of the simplicial set whose k-simplices consist of (k + 1)-tupels (g0, g1, . .,.gk) of elements gi in G. This also a model for E_G* * [1, Example 2.6]. The spaces XG and P1 (G) have the same underlying sets but in general they have different topologies. The identity map induces a (continuous) G-map P1 (G) ! XG which is a G-homotopy equivalence, but in general not a G-homeomorphism (see also [75, 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=G0 is compact for G0 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 J_G. 17 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 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 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. Sup- pose 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 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 E_G. Proof.By [13, Corollary II.2.8 on page 179] the K-fixed point set of X is non- empty convex subset of X and hence contractible for any compact subgroup_ K G. |__| 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 2 G and each open cell e with g . e \ e 6= ; we have gx = x for any x 2 e. Assume that the isotropy group of each x 2 T is compact. Then T is a 1-dimensional model for ECOMOP (G) = JCOMOP (G). Proof.We conclude from Remark 1.3 that T is a G-CW -complex and all isotropy groups are compact open. Let H G be compact open. If e0 is a zero-cell in T , then H . e0 is a compact discrete set and hence finite. Let T 0be 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 0that T 0has a vertex which is fixed under the H-action (see [65, page 20] or [25, Proposition 4.7 on page 17]). Hence T H is non-empty. If e and f are vertices 18 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 ECOMOP (G) which is the_same as JCOMOP (G) by Theorem 3.7 (i). |__| Let G be a locally compact Hausdorff group. Suppose that G acts continu- ously on a tree T such that for each element g 2 G and each open cell e with g . e \ e 6= ; we have gx = x for any x 2 e. If the G-action on a tree has possibly not compact isotropy groups, one can nevertheless get nice models for ECOMOP (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 ` q ` e2E G=He?x {-1, 1}----! T0 = v2VG=Kv? ?y ?y (4.8) ` e2EG=He x [-1, 1]----! T where the left vertical arrow is the obvious inclusion. Fix e 2 E and oe 2 {-1,* * 1}. Choose elements v(e, oe) 2 V and g(e, oe) 2 G such that q restricted to G=He x {oe} is the G-map G=He ! G=Kv(e,oe)which sends 1He to g(e, oe)Kv(e,oe). Then conjugation with g(e, oe) induces a group homomorphism cg(e,oe):He ! Kv(e, 1) and there is an up to equivariant homotopy unique cg(e,oe)-equivariant cellular map fg(e,oe):ECOMOP (He) ! ECOMOP (Ke(g,oe)). Define a G-map a a Q: G xHe ECOMOP (He) x {-1, 1} ! G xKv ECOMOP (Kv) e2E v2V by requiring that the restriction of Q to GxHe ECOMOP (He)x{oe} is the G-map G xHe ECOMOP (He) ! G xKv(e,oe)ECOMOP (Kg(e,oe)), (g, x) 7! (g, fg(e,oe)(x)). Let TCOMOP be the G-pushout ` Q ` e2E G xHe ECOMOP?(He) x {-1, 1}----! v2V G xKv?ECOMOP (Kv) ?y ?y ` e2EG xHe ECOMOP (He) x [-1, 1]----! TCOMOP The G-space TCOMOP 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 ECOMOP (L) the restriction resLGECOMOP (G) of ECOMOP (G) to L and ~= that there is a G-homeomorphism GxL resLGECOMOP (G) -! G=LxECOMOP (G) which sends (g, x) to (gL, gx). This implies that TCOMOP is G-homotopy equiv- alent to T xECOMOP (G) with the diagonal G-action. If H G is compact open, then T H is contractible. Hence (T x ECOMOP (G))H is contractible for compact open subgroup H G. Theorem 1.9 (ii) shows 19 Theorem 4.9 (Models based on actions on trees). The G-CW -complex TCOMOP is a model for ECOMOP (G). The point is that it may be possible to choose nice models for the various spaces ECOMOP (He) and ECOMOP (Kv) and thus get a nice model for ECOMOP (G). If all isotropy groups of the G-action on T are compact, we can choose all spac* *es ECOMOP (He) and ECOMOP (Kv) 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 x {-1, 1} ! V ) consisting of two sets V and E and a map s. The associated oriented graph is the pushout E x {-1, 1} --s--! V ?? ? y ?y E x [0, 1]----! X So V is the set of vertices, E the set of edges, and for a edge e 2 E its initi* *al 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 {Kv | v 2 V } and {He | e 2 E} with V and E as index sets together with injective group homomorphisms OEv,oe:He ! Ks(e,oe)for each e 2 E. Let X0 X be some maximal tree. We can associate to these data the fundamental group ß = ß(G, X, X0) as follows. Generators of ß are the elements in Kv for each v 2 V and the set {te | e 2 E}. The relations are the relations in each group Kv for each v 2 V , the relation te = 1 for e 2 V if e belongs to X0, and for each e 2 E and h 2 He we require t-1eOEe,-1(h)te = OEe,+1(h). It turns out that the obvious map Kv ! ß is an injective group homomorphism for each v 2 V and we will identify in the sequel Kv with its image in ß [25, Corollary 7.5 on page 33], [65, Corollary 1 in 5.2 on page 45]. We can assign to these data a tree T = T (X, X0, G) with ß-action as follows. Define a ß-map a a q : ß= im(OEe,-1) x {-1, 1} ! ß=Kv e2E v2V by requiring that its restriction to ß= im(OEe,-1)x{-1} is the ß-map given by t* *he projection ß= im(OEe,-1) ! ß=Ks(e,-1)and its restriction to ß= im(OEe,-1) x {1} is the ß-map ß= im(OEe,-1) ! ß=Ks(e,1)which sends g im(OEe,-1) to gteim (OEe,1). Now define a 1-dimensional G-CW -complex T = T (G, X, X0) using this ß-map q and the ß-pushout analogous to (4.8). It turns out that T is contractible [25, Theorem 7.6 on page 33], [65, 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 x {-1, 1} ! V sends s(e, oe) to the vertex for which q(G=Hex {oe}) meets and hence is equal to G=Ks(e,oe). Moreover, we get a graph of groups G on X as follows. Let {Kv | v 2 V } and {He | e 2 E} be the set of groups given by (4.8). 20 Choose an element g(e, oe) 2 G such that the G-map induced by q from G=He to G=Ks(e,oe)sends 1He to g(e, oe)Ks(e,oe). Then conjugation with g(e, oe) induces* * a group homomorphism OEe,oe:He ! Ks(e,oe). After a choice of a maximal tree X0 in X one obtains an isomorphism G ~=ß(G, X, X0). (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 [65, x5]. Example 4.10 (The graph associated to amalgamated products). Con- sider the graph D with one edge e and two vertices v-1 and v1 and the map s: {e} x {-1, 1} ! {v-1, v1} which sends (e, oe) to voe. 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 He and groups K-1 and K1 together with injective group homomorphisms OEoe:He ! Koefor oe 2 {-1, 1}. There is only one choice of a maximal subtree in D, namely D it- self. Then the fundamental group ß of this graph of groups is the amalgamated product of K-1 and K1 over He with respect to OE-1 and OE1, i.e. the pushout of groups He -OE-1---!K-1 ? ? OE1?y ?y K1 ----! ß Choose OEoe-equivariant maps foe:E_He ! E_Koe. They induce ß-maps Foe:ß xHe E_He ! ß xKoeE_Koe, (g, x) 7! (g, foe(x)). We get a model for E_ß as the ß-pushout ` F ` ß xHe E_He x {-1, 1}F-1---1---!ß xK-1 E_K-1 ß xK1 E_K1 ?? ? y ?y ß xHe E_He x [-1, 1]----! E_ß 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} x {-1, 1} ! {v}. Of course this graph is homeomorphic to S1. Let G be a graph of groups on S. It consists of two groups He and Kv and two injective group homomorphisms OEoe:He ! Kv for oe 2 {-1, 1}. There is only one choice of a maximal subtree, namely {v}. The fundamental group ß of G is the so called HNN-extension associated to the data OEoe:He ! Kv for oe 2 {-1, 1}, i.e. the group generated by the elements of Kv and a letter tv whose relations are those of Kv and the relations t-1vOE-1(h)tv = OE1(h) for all h 2 He. Recall that the natural map Kv ! ß is injective and we will identify Kv with its image in ß. Choose OEoe-equivariant maps foe:E_He ! E_Kv. Let Foe:ß xOE-1E_He ! ß x E_Kv be the ß-map which sends (g, x) to gf-1(x) for 21 oe = -1 and to gtef1(x) for oe = 1. Then a model for E_ß is given by the ß-pushout ` ß xOE-1E_He x {-1, 1}F-1--F1----!ß xKv E_Kv ?? ? y ?y ß xOE-1E_He x [-1, 1]----! E_ß Notice that this looks like a telescope construction which is infinite to both sides. Consider the special case, where He = Kv, OE-1 = id and OE1 is an automorphism. Then ß is the semidirect product Kv oOE1Z. Choose a OE1- equivariant map f1: E_Kv ! E_Kv. Then a model for E_ß is given by the to both side infinite mapping telescope of f1 with the Kv oOE1Z action, for which Z acts by shifting to the right and k 2 Kv acts on the part belonging to n 2 Z by multiplication with OEn1(k). If we additionally assume that OE1 = id, then ß = Kv x Z and we get E_Kv x R as model for E_ß. Remark 4.12. All these constructions yield also models for EG = ETR (G) if one replaces everywhere the spaces E_He and E_Kv by the spaces EHe and EKv. 4.6 Affine Buildings Let 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 are contained in some common apartment; (iii)If two apartments both contain two simplices A and B of , 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 call* *ed 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: x ! [0, 1) which is non-positively curved and complete. The building with this metric is a CAT(0)-space. A simplicial automorphism of 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. I* *t is the set of points {z 2 | d(x, y) = d(x, z) + d(z, y)}. For x, y 2 and t 2 [* *0, 1] let tx + (1 - t)y be the point z 2 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 : x x [0, 1] ! , (x, y, t) 7! tx + (1 - t)y is continuous. This implies that is contractible. All these facts are taken f* *rom [17, Section 3 in Chapter VI] and [13, Theorem 10A.4 on page 344]. 22 Suppose that the group G acts on by isometries. If G maps a non-empty bounded subset A of to itself, then the G-action has a fixed point [17, Theor* *em 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 2 G also the segment [x, y] must lie* * in G and hence the map r above induces a continuous map G x G x [0, 1] ! 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 auto- morphisms such that each isotropy group is compact. Then each isotropy group is compact open, is a model for JCOMOP (G) and the barycentric subdivision 0 is a model for both JCOMOP (G) and ECOMOP (G). If we additionally assume that G is totally disconnected or is a Lie group, then is a model for both J_G and 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 fi(G) [70],[71]. Then fi(G) is a model for J_G and fi(G)0is a model for E_G by Theorem 4.13. 4.7 The Rips Complex of a Word-Hyperbolic Group Let G be a finitely generated discrete group. Let S be a finite set of generato* *rs. We will always assume that S is symmetric, i.e. that the identity element 1 2 G does not belong to S and s 2 S implies s-1 2 S. For g1, g2 2 G let dS(g1, g2) be the minimal natural number n such that g-11g2 can be written as a word s1s2. .s.n. This defines a left G-invariant metric on G, the so called word metric. A metric space X = (X, d) is called ffi-hyperbolic for a given real number ffi 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)}4+.2ffi.15) A group G with a finite symmetric set S of generators is called ffi-hyperbol* *ic if the metric space (G, dS) is ffi-hyperbolic. The Rips complex Pd(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 (g0, g1, . .g.k) of pai* *rwise distinct elements gi 2 G satisfying dS(gi, gj) d for all i, j 2 {0, 1, . .,.k* *}. The obvious G-action by simplicial automorphisms on Pd(G, S) induces a G- action by simplicial automorphisms on the barycentric subdivision Pd(G, S)0 (see Example 1.5). The following result is proved in [56], [57]. Theorem 4.16 (Rips complex). Let G be a (discrete) group with a finite symmetric set of generators. Suppose that (G, S) is ffi-hyperbolic for the real number ffi 0. Let d be a natural number with d 16ffi +8. Then the barycentr* *ic subdivision of the Rips complex Pd(G, S)0 is a finite G-CW -model for E_G. 23 A metric space is called hyperbolic if it is ffi-hyperbolic for some real nu* *mber ffi 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, dS) is a hyperbo* *lic metric space. Since for metric spaces the property hyperbolic is invariant under quasiisometry and for two symmetric finite sets S1 and S2 of generators of G the metric spaces (G, dS1) and (G, dS2) 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 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 E_G. Namely, let G(R) be the R-points of a semisimple Q- group 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 EFIN (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], [64]) is a finite A-CW -model for EFIN (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 Fn be the free group of rank n. Denote by Out(Fn) the group of outer automorphisms of Fn, i.e. the quotient of the group of all automorphisms of Fn by the normal subgroup of inner automorphisms. Culler and Vogtmann [21], [76] have constructed a space Xn called outer space on which Out(Fn) acts with finite isotropy groups. It is analogous to the Teichmüller space of a surface w* *ith the action of the mapping class group of the surface. Fix a graph Rn with one vertex v and n-edges and identify Fn with ß1(Rn, v). A marked metric graph (g, ) consists of a graph with all vertices of valence at least three, a hom* *otopy equivalence g :Rn ! called marking and to every edge of there is assigned a positive length which makes into a metric space by the path metric. We call two marked metric graphs (g, ) and (g0, 0) equivalent of there is a homothety h: ! 0such that g O h and h0are homotopic. Homothety means that there is a constant ~ > 0 with d(h(x), h(y)) = ~ . d(x, y) for all x, y. Elements in outer space Xn are equivalence classes of marked graphs. The main result in [21] is that X is contractible. Actually, for each finite subgroup H Out(Fn) the H-fixed point set XHn is contractible [43, Propostion 3.3 and Theorem 8.1], [78, Theorem 5.1]. The space Xn contains a spine Kn which is an Out(Fn)-equivariant deforma- tion retraction. This space Kn is a simplicial complex of dimension (2n - 3) on which the Out(Fn)-action is by simplicial automorphisms and cocompact. Ac- tually the group of simplicial automorphisms of Kn is Out(Fn) [14]. Hence the 24 barycentric subdivision K0nis a finite (2n - 3)-dimensional model of E_Out(Fn). 4.10 Mapping Class groups Let sg,rbe 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 Fg ! Fg, 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 Teichmül- ler space Tgs,ris a contractible space on which sg,racts properly. Actually T* *gs,r is a model for EFIN ( sg,r) by the results of Kerckhoff [41]. We could not find a clear reference in the literature for the to experts kn* *own statement that there exist a finite sg,r-CW -model for EFIN ( sg,r). The work* * of Harer [32] on the existence of a spine and the construction of the spaces TS(f* *fl)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 2 MFIN , M 6= {1} ) NG M = M; For such a group there is a nice model for E_G with as few non-free cells as possible. Let {(Mi) | i 2 I} be the set of conjugacy classes of maximal finite subgroups of Mi` Q. By attaching free G-cells we get an inclusion of G-CW - complexes j1: i2IG xMi EMi ! EG, where EG is the same as ETR (G), i.e. a contractible free G-CW -complex. Define E_G as the G-pushout ` j1 i2IG xMi?EMi ----! EG? u1?y ?yf1,EG (4.17) ` i2IG=Mi ----!k1E_G where u1 is the obvious G-map obtained by collapsing each EMi to a point. We have to explain why 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 fin* *ite. We have to show for H G finite that (E_G)H contractible. We begin with the case H 6= {1}. Because of conditions (M) and (NM) there is precisely one index 25 i0 2 I such that H is subconjugated to Mi0and is not subconjugated to Mi for i 6= i0 and we get _ ! H a H G=Mi = (G=Mi0) = {pt.}. i2I Hence E_GH = {pt.}. It remains to treat H = {1}. Since u1 is a non-equivariant homotopy equivalence and j1 is a cofibration, f1 is a non-equivariant homotopy equivalence and hence E_G is contractible (after forgetting the group action). Here are some examples of groups Q which satisfy conditions (M) and (NM): o Extensions 1 ! Zn ! G ! F ! 1 for finite F such that the conjugation action of F on Zn is free outside 0 2 Zn. The conditions (M), (NM) are satisfied by [54, Lemma 6.3]. o Fuchsian groups F The conditions (M), (NM) are satisfied (see for instance [54, Lemma 4.5]). In [54] the larger class of cocompact planar groups (sometimes also called cocompact NEC-groups) is treated. o One-relator groups G Let G be a one-relator group. Let G = <(qi)i2I| 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 {qi| i 2 I}. Then r is an element in F . There exists an element s 2 F and an integer m 2 such that r = sm , the cyclic subgroup C generated by the class _s2 G represented by s has order m, any finite subgroup of G is subconjugated to C and for any g 2 G the implication g-1Cg \ C 6= 1 ) g 2 C holds. These claims follows from [55, 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 = <(qi)i2I | r> be a presentation with one relation. There is up to conjugacy one maximal finite subgroup C which is cyclic. Let p: *i2IZ ! G be the epimorphism from the free groupWgenerated by the set I to G, which sends the generator i 2 I to qi. Let Y ! i2IS1 be the G-covering associated to the epimorphism p. There is a 1-dimensional unitary C-representation V and a C-map f :SV ! resCGY 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 E_G by the G-pushout _f G xC SV ----! Y ?? ? y ?y G xC DV ----! E_G 26 __ where f sends (g, x) to gf(x). Thus we get a 2-dimensional G-CW -model for E_G such that 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\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 [55, Chapter III xx9 -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 SL2(R) and SL2(Z). Let H2 be the 2-dimensional hyperbolic space. We will use either the upper half-plane model or the Poincar'e disk model. The group SL2(R) acts by isomet- ric diffeomorphisms`on'the upper half-plane by Moebius transformations, i.e. a matrix ac bd acts by sending a complex number z with positive imaginary part to az+b_cz+d. This action is proper and transitive. The isotropy group of * *z = i is SO(2). Since H2 is a simply-connected Riemannian manifold, whose sec- tional curvature is constant -1, the SL2(R)-space H2 is a model for E_SL2(R) by Theorem 4.5. One easily checks that SL2(R) is a connected Lie group and SO(2) SL2(R) is a maximal compact subgroup. Hence SL2(R)=SO(2) is a model for E_SL2(R) by Theorem 4.3. Since the SL2(R)-action on H2 is transitive and SO(2) is the isotropy group at i 2 H2, we see that the SL2(R)-manifolds SL2(R)=SO(2) and H2 are SL2(R)-diffeomorphic. Since SL2(Z) is a discrete subgroup of SL2(R), the space H2 with the obvious SL2(Z)-action is a model for E_SL2(Z) (see Theorem 4.4). The group SL2(Z) is isomorphic to the amalgamated product Z=4 *Z=2Z=6. From Example 4.10 we conclude that a model for E_SL2(Z) is given by the following SL2(Z)-pushout ` F ` SL2(Z)=(Z=2) x {-1, 1}-F-1---1--!SL2(Z)=(Z=4) SL2(Z)=(Z=6) ?? ? y ?y SL2(Z)=(Z=2) x [-1, 1]----! E_SL2(Z) where F-1 and F1 are the obvious projections. This model for E_SL2(Z) is a tree, which has alternately two and three edges emanating from each vertex. The other model H2 is a manifold. These two models must be SL2(Z)-homotopy equivalent. They can explicitly be related by the following construction. Divide the Poincar'e disk into fundamental domains for the SL2(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 Poincar'e disk, 27 is a tree T with SL2(Z)-action. This is the tree model above. The tree is a SL2(Z)-equivariant deformation retraction of the Poincar'e disk. A retraction is given by moving a point p in the Poincar'e 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 SL2(Qbp) and hence is a model for E_SL2(Qbp) (see [17, page 134]). Since SL2(Z) is a discrete subgroup of SL2(Qbp), we get another reason why this tree is a model for SL2(Z). 4.14 Manifold Models It is an interesting question, whether one can find a model for E_G which is a smooth G-manifold. One may also ask whether such a manifold model realizes the minimal dimension for E_G or whether the action is cocompact. Theorem 4.5 gives some information about these questions for simply connected non- positively curved Riemannian manifolds and Theorem 5.24 for discrete sub- groups 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 co- compactly on a contractible manifold (without boundary), but there is no finite G-CW -model for E_G [23, Theorem 1.1]. 28 5 Finiteness Conditions In this section we investigate whether there are models for EF (G) which satisfy certain finiteness conditions such as being finite, being of finite type or bei* *ng 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 RG- module M is an RG-chain complex P* of projective or free respectively RG- modules with Pi = 0 for i -1 such that Hi(P*) = 0 for i 1 and H0(P*) is RG-isomorphic to M. If additionally each RG-module Pi 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 conditi* *on that for any RG-module N we have ExtiRG(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 FPn, if it admits a projective RG-resolution P* such that Pi is finitely generated for i n and of type FP1 if it admits a projective RG-resolution P* such that Pi is finitely generated for all i. A group G is of type FPn or FP1 respectively if the trivi* *al ZG-module Z is of type FPn or FP1 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 RG- resolution 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 FPn 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 FPn; (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 FP1 ; 29 (e)There exists groups G which are of type FP2 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\gBG = C\EG is a finite dimensional model for BC. Hence there is an integer d such that we have Hi(BC) = 0 for i d. This implies that C is trivial [16, (2.1) in II.3 on page 35]. Hence G is torsi* *onfree. (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 [67] and [69]. (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 OrF (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 mapG (G=H, G=K) 30 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 ZOrF (G)(Z mapG (G=?, G=K), N) -! N(G=K), OE 7! OE(G=K)(idG=K). This is a direct consequence of theLYoneda Lemma. A ZOr F(G)-module is free if it is isomorphic to a direct sum i2IZ map(G=?, G=Ki) for appropriate choice of objects G=Kiand index set I. A ZOr F(G)-module is calledLfinitely generated if it is a quotient of a ZOr F(G)-module of the shape i2IZ map(G=?, G=Ki) with a finite index set I. Notice that a lot of standard facts for Z-modules ca* *rry 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 summan* *d 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 FP1 carry directly over. Each ZOr F(G)-module has a projective resolution. The trivial ZOr F(G)- module Z_is the constant functor from OrF (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 [47, Section 9]. The next result is proved in [50, 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*(EF (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 EF (G) if and only if the trivial ZOr F(G)-module Z_has cohomological dimension d; (ii)There is a G-CW -model for EF (G) of finite type if and only if EF (G) has a G-CW -model with finite 2-skeleton and the trivial ZOr F(G)-module Z_ is of type FP1 ; (iii)There is a finite G-CW -model for EF (G) if and only if EF (G) has a G- CW -model with finite 2-skeleton and the trivial ZOr F(G)-module Z_has a finite free resolution over OrF (G); (iv)There is a G-CW -model with finite 2-skeleton for E_G = EFIN (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 WG H := NG H=H is finitely presented. In the case, where we take F to be the trivial family, Theorem 5.2 (i) reduc* *es 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. 31 Remark 5.3. Nucinkis [63] investigates the notion of FIN -cohomological di- mension and relates it to the question whether there are finite-dimensional mod- ules for E_G. It gives another lower bound for the dimension of a model for E_G but is not sharp in general [11]. 5.3 Reduction from Topological Groups to Discrete Groups The discretization Gd 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 Fd the same set of subgroups, but now in connection with Gd. Notice that Fd is again a family. We will need the following condition For any closed subgroup H G the projection p: G ! G=H has (S) a local cross section, i.e. there is a neighborhood U of eH together with a map s: U ! G satisfying p O s = idU. 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 cov- ering dimension [61]. The metric needed in [61] 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 [50, 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 EF (G) that is d-dimensional or finite or of finite type respectively if and only if t* *here is a Gd-CW -model for EFd(Gd) 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 G := G=G0. Then there is a G-CW -model for E_G that_is_d-dimensional_or finite or of finite type respectively if and only if E_G has a 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 COM the family of compact subgroups of its component group G and let d 3. Then (i)There is a d-dimensional_G-CW -model for E_G if and only if the trivial ZOr ____COMd(G d)-module Z_has cohomological dimension d; __ (ii)There_is a G-CW -model for G_of finite type if and only if E____COMd(G_d) * *has a G d-CW -model with finite 2-skeleton and the trivial ZOr ____COMd(G d)-mod* *ule Z_is of type FP1 ; 32 __ (iii)There_is a finite G-CW -model for E_G if and only if E____COMd(G d)_has a G d-CW -model with finite 2-skeleton and the trivial ZOr ____COMd(G d)-mod* *ule Z_has a finite free resolution. In particular we see from Theorem 5.5 that, for a Lie group G, type questions about E_G_are equivalent to the corresponding type_questions_of E_ß0(G), since ß0(G) = G is discrete. In this case the family COM d appearing in Theorem 5.6. is just the family FIN of finite subgroups of ß0(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 6= 1}. (5.7) An element g 2 G sends K to gKg-1 and the poset-structure comes from inclu- sion 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 NG H be the normalizer and let WG H := NG H=H be the Weyl group of H G. Notice for a G-space X that XH inherits a WG 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=KH is a finite union of WG H-orbits of the shape WG H=L for finite L WG H. Now one easily checks Lemma 5.8. The WG H-space E_GH is a WG H-CW -model for E_WG H. In par- ticular, if E_G has a G-CW -model which is finite, of finite type or d-dimensio* *nal respectively, then there is a WG H-model for E_WG 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(<1) 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 re- spectively. A group G satisfies the condition B(d) if WG H satisfies the condit* *ion b(d) for any finite subgroup H G. The length l(H) 2 {0, 1, . .}.of a finite group H is the supremum over all p for which there is a nested sequence H0 H1 . . .Hp of subgroups Hi of H with Hi6= Hi+1. Lemma 5.10. Suppose that there is a d-dimensional G-CW -complex X with finite isotropy groups such that Hp(X; Z) = Hp(*, Z) for all p 0 holds. This assumption is for instance satisfied if there is a d-dimensional G-CW -model for E_G. Then G satisfies condition B(d). 33 Proof. Let H G be finite. Then X=H satisfies Hp(X=H; Z) = Hp(*, Z) for all p 0 [12, III.5.4 on page 131]. Let C* be the cellular ZWG H-chain complex of X=H. This is a d-dimensional resolution of the trivial ZWG H-module Z and each chain module is a sum of ZWG H-modules of the shape Z[WG H=K] for some finite subgroup K WG H. Let N be a ZWG H-module such that N is projective over ZK for any finite subgroup K WG H. Then C* Z N with the diagonal WG H-operation is a d-dimensional projective ZWG 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 E_G such that for any finite subgroup H G dim (E_GH ) = d(H); (ii)We have for any finite subgroup H G and for any ZWG H-module M Hd(H)+1ZWGH(EWG H x (C|P(WG H)|, |P(WG H)|); M) = 0; (iii)We have for any finite subgroup H G that its Weyl group WG H satisfies b(< 1) and that there is a subgroup (H) WG H of finite index such that for any Z (H)-module M Hd(H)+1Z((H)E (H) x (C|P(WG H)|, |P(WG H)|); M) = 0. Then (i) implies both (ii) and (iii). If there is an upper bound on the leng* *th 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 [48, 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. 34 Remark 5.13. If WG H contains a non-trivial normal finite subgroup L, then |P(WG H)| is contractible and Hd(H)+1ZWGH(EWG H x (C|P(WG H)|, |P(WG H)|); M)= 0; Hd(H)+1Z((H)E (H) x (C|P(WG H)|, |P(WG H)|); M)= 0. The proof of this fact is given in [48, Example 1.8]. The next result is taken from [48, Theorem 1.10]. A weaker version of it for certain classes of groups and in l exponential dimension estimate can be found in [42, Theorem B] (see [48, 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 E_G such that for any finite subgroup H G dim(E_GH ) max{3, d} + (l - l(H))(d + 1) holds. In particular 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 ! ! G ! ß ! 1. We want to investigate whether finiteness conditions about the type of a classifying space for FIN for and ß carry over to the one of G. The proof of the next Theorem 5.15 is taken from [48, 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 sub- groups of ß. Suppose that E_ has a k-dimensional -CW -model and E_ß has a m-dimensional ß-CW -model. Then E_G has a (dk + m)-dimensional G-CW - model. Theorem 5.16. Suppose that has the property that for any group which contains as subgroup of finite index, there is a k-dimensional -CW -model for E_ . Suppose that E_ß has a m-dimensional ß-CW -model. Then E_G has a (k + m)-dimensional G-CW -model. We will see in Example 5.26 that the condition about in Theorem 5.16 is automatically satisfied if is virtually poly-cyclic. The next two results are taken from [48, Theorem 3.2 and Theorem 3.3]). Theorem 5.17. Suppose for any finite subgroup ß0 ß and any extension 1 ! ! 0! ß0! 1 that E_ 0has a finite 0-CW -model or a 0-CW -model of finite type respectively and suppose that E_ß has a finite ß-CW -model or a ß-CW -model of finite type respectively. Then E_G has a finite G-CW -model or a G-CW -model of finite type respectively. 35 Theorem 5.18. Suppose that is word-hyperbolic or virtually poly-cyclic. Sup- pose that E_ß has a finite ß-CW -model or a ß-CW -model of finite type respec- tively. Then E_G has a finite G-CW -model or a G-CW -model of finite type respectively. 5.6 One-Dimensional Models for 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 dis- crete group. Then there exists a 1-dimensional model for 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 E_G if a* *nd only if G contains a finitely generated free subgroup of finite index [40, Theo* *rem 1]. If G is torsionfree, we rediscover the results due to Swan and Stallings st* *ated 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 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 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 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 of finite index such that has cohomological dimension d. This is independent of the choice of G because for two torsionfree subgroups , 0 G we have that has cohomological dimension d if and only if 0 has cohomological dimension d. The next two results are taken from [48, Lemma 6.1, Theorem 6.3, Theorem 6.4]. Lemma 5.21. If G satisfies b(d) or B(d) respectively, then any subgroup 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 of finite index, then the followi* *ng assertions are equivalent: (i)G satisfies B(d); 36 (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(E_G). Let G be a discrete group which contains a torsionfree subgroup of finite index and h* *as 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(E_G) for any model for E_G and there is a G-CW -model for E_G such that for any finite subgroup H G dim (E_GH ) = max{3, d} + l - l(H) holds. In particular there exists a model for 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 sub- group 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 E_G. Suppose additionally that G contains a torsionfree subgroup 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 E_G. The restriction of E_G to is a -CW -model for E_ and hence \E_G is a CW -model for B . This implies vcd(G) := cd( ) dim(L=K). Obviously \L=K is a manifold without boundary. Suppose that \L=K is compact. Then \L=K is a closed manifold and hence its homology with Z=2-coefficients in the top dimension is non-trivial. This implies cd( ) dim( \L=K) and hence vcd(G) = dim (L=K). If \L=K is not compact, it contains a CW - complex X \L=K of dimension smaller than \L=K such that the inclusion of X into \L=K is a homotopy equivalence. Hence X is another model for_ B . This implies cd( ) < dim(L=K) and hence vcd(G) < dim(L=K). |__| Remark 5.25. An often useful strategy to find smaller models for EF (G) is to look for a G-CW -subcomplex X EF (G) such that there exists a G-retraction r :EF (G) ! X, i.e. a G-map r with r|X = idX. Then X is automatically another model for EF (G). We have seen this already in the case SL2(Z), where we found a tree inside H2 = SL2(R)=SO(2) as explained in Subsection 4.13. This method can be used to construct a model for E_SLn(Z) of dimension n(n-1)_2 and to show that the virtual cohomological dimension of SLn(Z) is n(n-1)_2. Notice that SLn(R)=SO(n) is also a model for E_SLn(Z) by Theorem 4.4 but has dimension n(n+1)_2- 1. 37 Example 5.26 (Virtually poly-cyclic groups). Let the group be virtually poly-cyclic, i.e. contains a subgroup 0 of finite index for which there is a finite sequence {1} = 00 01 . . . 0n= 0 of subgroups such that 0i-1 is normal in 0iwith cyclic quotient 0i= 0i-1for i = 1, 2, . .,.n. Denote by r the number of elements i 2 {1, 2, . .,.n} with 0i= 0i-1~=Z. The number r is called the Hirsch rank. The group contains a torsionfree subgroup of finite index. We call 0poly-Z if r = n, i.e. all quotients 0i= 0i-1are infinite cycl* *ic. We want to show: (i)r = vcd( ); (ii)r = max {i | Hi( 0; Z=2) 6= 0} for one (and hence all) poly-Z subgroup 0 of finite index; (iii)There exists a finite r-dimensional model for E_ and for any model E_ we have dim(E_ ) r. We use induction over the number r. If r = 0, then 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 ! 0 ! p-!V ! 1 for some virtually poly-cyclic group 0 with r( 0) = r( ) - 1 and some group V which contains Z as subgroup of finite index. The induction hypothesis applies to any group which contains 0 as subgroup of finite index. Since V maps surjectively to Z or the infinite dihedral group D1 with finite kernel and both Z and D1 have 1-dimensional models for their classifying space for proper group actions, there is a 1-dimensional model for E_V . We conclude from Theorem 5.16 that there is a r-dimensional model for E_ . The existence of a r-dimensional model for E_ implies vcd( ) r. For any torsionfree subgroup 0 of finite index we have max {i | Hi( 0; Z=2) 6= 0} vcd( ), It is not hard to check by induction over r that we can find a sequence of torsionfree subgroups {1} 0 1 . . . r such that i-1 is normal in i with i= i-1~= Z for i 2 {1, 2, . .,.r} and r has finite index in . We show by induction over i that Hi( 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 foll* *ows from the part of the long exact Wang sequence Hi( i-1; Z=2) = 0 ! Hi( i; Z=2) ! Hi-1( i-1; Z=2) = Z=2 -id-Hi-1(f;Z=2)-=-0----------!H i-1( i-1; Z=2) which comes from the Hochschild-Serre spectral sequence associated to the ex- tension 1 ! i-1 ! i ! Z ! 1 for f : i-1 ! i-1 the automorphism induced by conjugation with some preimage in i of the generator of Z. This implies r = max{i | Hi( r; Z=2) 6= 0} = cd( r) = vcd( ). Now the claim follows. 38 The existence of a r-dimensional model for E_G is proved for finitely gen- erated nilpotent groups with vcd(G) r for r 6= 2 in [62], where also not necessarily finitely generated nilpotent groups are studied. The work of Dekimpe-Igodt [24] or Wilking [80, Theorem 3] implies that there is a model for EFIN ( ) whose underlying space is Rr. 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 sub- group of finite index and has virtual cohomological dimension d, does there exist a d-dimensional G-CW -model for E_G? The following four problems for discrete groups G are stated in the problem lists appearing in [48] and [77]. Problem 5.28. Let H G be a subgroup of finite index. Suppose that E_H has a H-CW -model of finite type or a finite H-CW -model respectively. Does then 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 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 BWG H of finite type for any finite subgroup H G? Problem 5.31. Let 1 ! -i!G -p!ß ! 1 be an exact sequence of groups. Suppose that there is a -CW -model of finite type for E_ and a G-CW -model of finite type for E_G. Is then there a ß-CW -model of finite type for E_ß? Leary and Nucinkis [46] 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 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 CG H of H in G is not of type FP1 ; 39 (iii)There exists a group G of type VF which contains infinitely many conju- gacy classes of finite subgroups; (iv)There exists an extension 1 ! ! G ! ß ! 1 such that E_ and E_G have finite G-CW -models but there is no G-CW -model for E_ß of finite type. 40 6 The Orbit Space of 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\E_G of E_G. The following result shows that this is a difficult problem in general and can only be solved in special c* *ases. It was proved by Leary and Nucinkis [45] based on ideas due to Baumslag-Dyer- Heller [8] and Kan and Thurston [39]. Theorem 6.1 (The homotopy type of G\E_G). Let X be a connected CW - complex. Then there exists a group G such that G\E_G is homotopy equivalent to X. There are some cases, where the quotient G\E_G has been determined ex- plicitly using geometric input. We mention a few examples. (i)Let G be a planar group (sometimes also called NEC) group, i.e. a dis- continuous group of isometries of the two-sphere S2, the Euclidean plane R2, or the hyperbolic plane H2. Examples are Fuchsian groups and two- dimensional crystallographic groups. If G acts on R2 or H2 and the action is cocompact, then R2 or H2 is a model for E_G and the quotient space G\E_G is a compact 2-dimensional surface. The number of boundary com- ponents, its genus and the answer to the question, whether G\E_G is ori- entable, can be read off from an explicit presentation of G. A summary of these details can be found in [54, Section 4], where further references to papers containing proofs of the stated facts are given; (ii)Let G = <(qi)i2I | r> be a one-relator group. Let F be the free group on the letters {qi | i 2 I}. Then r is an element in F . There exists an element s 2 F and an integer m 1 such that r = sm , the cyclic subgroup C generated by the class _s2 G represented by s has order m, any finite subgroup of G is subconjugated to C and for any g 2 G the implication g-1Cg \ C 6= {1} ) g 2 C holds (see [55, 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 E_G described in Sub- section 4.12. Let us compute the integral homology of BG and G\E_G. Since G\E_G has precisely one 2-cell and is two-dimensional, H2(G\E_G) is either trivial or infinite cyclic and Hk(G\E_G) = 0 for k 3. We obtain the short exact sequence 0 ! H2(BG) H2(q)----!H2(G\E_G)@2-!H1(BC) H1(Bi)-----!H1(BG) H1(q)----!H 1(G\E_G) ! 0 and for k 3 isomorphisms ~= Hk(Bi): Hk(BC) -! Hk(BG) 41 from the pushout coming from (4.17) BC ---i-! BG ?? ? y ?y {pt.}----! G\E_G Hence H2(G\E_G) = 0 and the sequence 0 ! H1(BC) H1(Bi)-----!H1(BG) H1(q)----!H1(G\E_G) ! 0 is exact, provided that H2(BG) = 0. Suppose that H2(BG) 6= 0. Hopf's Theorem says that H2(BG) ~=R \ [F, F ]=[F, R] if R is the subgroup of G normally generated by r 2 F (see [16, Theorem 5.3 in II.5 on page 42]). For every element in R \ [F, F ]=[F, R] there exists n 2 Z such that rn belongs to [F, F ] and the element is represented by rn. Hence there is n 1 such that rn does belong to [F, F ]. Since F=[F, F ] is torsion- free, also s and r belong to [F, F ]. We conclude that both H2(BG) and H2(G\E_G) are infinite cyclic groups, H1(BC) ! H1(BG) is trivial and ~= H1(q): H1(BG) -! H1(G\E_G) is bijective. We also see that H2(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 GL3(Z) consisting of upper triangular matrices with 1 on the diagonals. Consider the Z=4-action given by 0 1 0 1 1 x y 1 -z y - xz @ 0 1 z A 7! @ 0 1 x A . 0 0 1 0 0 1 Then a key result in [49] is that G\E_G is homeomorphic to S3 for G = HeioZ=4; (iv)A key result in [66, Corollary on page 8] implies that for G = SL3(Z) the quotient space G\E_G is contractible. 42 7 Relevance and Applications of Classifying Spa- ces for Families In this section we discuss some theoretical aspects which involve and rely on t* *he 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 equiv- ariant KK-theory due to Kasparov one can assign to a COM-numerable G-space X its equivariant K-theory KGn(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 Kp(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: KGn(J_G) -! Kn(Cr(G)) is bijective for all n 2 Z. More information about this conjecture and its relation and application to other conjectures and problems can be found for instance in [7], [34], [53], [5* *9], [75]. 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 HG*(X; K) graded over the integers and defined for G-CW -complexes X such that for any subgroup H G the abelian group HGn(G=H; K) is isomorphic to the algebraic K-groups Kn(RH) for n 2 Z. If R comes with an involution of rings, one can also construct a G-homology theory HG*(X; L<-1>) graded over the integers and defined for G-CW -complexes X such that for any subgroup H G the abelian group HGn(G=H; L<-1>) is isomorphic to the algebraic L-groups L-1n(RH) for n 2 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 Kn(RH) and the algebraic L-groups. 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 EVCYC(G) ! G=G asmb:HGn(EVCYC(G), K) ! HGn(G=G, K) = Kn(RG); (7.3) asmb:HGn(EVCYC(G), L-1 ) ! HGn(G=G, L-1 ) = L-1n(RG), (7.4) are bijective for all n 2 Z. 43 More information about this conjecture and its relation and application to other conjectures and problems can be found for instance in [28] and [53]. 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 HG*(X; Ktop) graded over the integers and defined for G-CW -complexes X such that for any subgroup H G the abelian group HGn(G=H; Ktop) is isomorphic to the topological K-groups Kn(C*r(H)) for n 2 Z and the assembly map appearing in the Baum-Connes Conjecture can be identified with the map induced by the projection J_G = 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 EVCYC(G) by 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 J_G, E_G or EVCYC(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 [52, Theorem 3.2]). It is a G-cohomology theory with a multiplicative structure. Assume that E_G has a finite G-CW - model. Let I K0G(E_G) be the augmentation ideal, i.e. the kernel of the map K0(E_G) ! Z sending the class of an equivariant complex vector bundle to its complex dimension. Let K*G(E_G)bIbe the I-adic completion of K*G(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 E_G. Then there is a canonical isomorphism ~ K*(BG) =-!K*G(E_G)bI. This result is proved in [52, Theorem 4.4], where a more general statement is given provided that there is a finite-dimensional model for 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]. 7.4 Classifying Spaces for Equivariant Bundles In [51] the equivariant K-theory for finite proper G-CW -complexes appearing in Subsection 7.3 above is extended to arbitrary proper G-CW -complexes (in- 44 cluding the multiplicative structure) using -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 L* *ie groups and certain families (see [73, Section 8 and 9 in Chapter I], [52, Lemma 2.4]). 7.5 Equivariant Homology and Cohomology Classifying spaces for families play a role in computations of equivariant ho- mology and cohomology for compact Lie groups such as equivariant bordism as explained in [72, Chapter 7], [73, Chapter III]. 45 8 Computations using Classifying Spaces for Fam- ilies 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 E_G and G\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 ! E_G be the up to G-homotopy unique G-map. Denote by ji:Mi! G the inclusion. 8.1 Group Homology We begin with the group homology Hn(BG) (with integer coefficients). Let Hep(X) be the reduced homology, i.e. the kernel of the map Hn(X) ! Hn({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 M L i2IHp(Bji) . .!.Hp+1(G\E_G)) @p+1---!Hep(BMi) ---------! Hp(BG) i2I -Hp(G\s)----!H @p p(G\E_G) -! . . .(8.1) In particular we obtain an isomorphism for p dim(E_G) + 2 M M ~= Hp(Bji): Hep(BMi) -! Hp(BG). (8.2) i2I i2I This example and the forthcoming ones show why it is important to get upper bounds on the dimension of E_G and to understand the quotient space G\E_G. For Fuchsian groups and for one-relator groups we have dim(G\E_G) 2 and it is easy to compute the homology of G\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 G- action look like M . .!.KGp+1(E_G) ! KGp(G xMi EMi) _ i2I ! M M ! KGp(G=Mi) KGp(EG) ! KGp(E_G) ! . . .(8.3) i2I 46 and M . .!.Kp+1(G\E_G) ! Kp(BMi) _ i2I ! M M M ! Kp({pt.}) Kp(BG) ! Kp(G\E_G) ! . . .(8.4) i2I i2I Notice that for a free G-CW -complex X there is a canonical isomorphisms KGp(X) ~=Kp(G\X). We can splice these sequences together and obtain the long exact sequence M M M . .!.Kp+1(G\E_G) ! KGp(G=Mi) ! Kp({pt.}) KGp(E_G) i2I i2I ! Kp(G\E_G) ! . . .(8.5) There are identification of KG0(G=Mi) with the complex representation ring RC(Mi) of the finite group Mi and of K0({pt.}) with Z. Under these iden- tification the map KG0(G=Mi) ! K0({pt.}) becomes the split surjective map ffl: RC(Mi) ! Z which sends the class of a complex Mi-representation V to the complex dimension of C C[Mi]V . The kernel of this map is denoted by eRC(Mi). The groups KG1(G=Mi) and K1({pt.}) vanish. The abelian group RC(Mi) and hence also eRC(Mi) are finitely generated free abelian groups. If Z Q is ring such that the order of any finite subgroup of G is invertible in , then t* *he map Z KGp(s): Z KGp(EG) ! Z KGp(G\E_G) is an isomorphism for all p 2 Z [54, Lemma 2.8 (a)]. Hence we conclude from the long exact sequence (8.5) 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 {(Mi) | i 2 I} be the set of conjugacy classes of maximal finite subgroups of G. Then there is an isomorphism ~= K1(C*r(G)) -! K1(G\E_G) and a short exact sequence M 0 ! eRC(Mi) ! K1(C*r(G)) ! K1(G\E_G) ! 0, i2I 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 computa- tion of the relevant groups Kn(RG) or L<-1>n(RG) respectively is equivalent to the computation of HGn(EVCYC(G), K) or HGn(EVCYC(G), L-1 ) respectively. The following result is due to Bartels [5]. Recall that E_G is the same as EFIN (G). 47 Theorem 8.7. (i) For every group G, every ring R and every n 2 Z the up to G-homotopy unique G-map f :EFIN (G) ! EVCYC(G) induces a split injection HGn(f; KR ): HGn(EFIN (G); KR ) ! HGn(EVCYC(G); KR ); (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 HGn(f; L<-1>R): HGn(EFIN (G); L<-1>R) ! HGn(EVCYC(G); L<-1>R). It remains to compute HGn(EFIN (G); K) and HGn(EVCYC(G), EFIN (G); K), if we arrange f to be a G-cofibration and think of EFIN (G) as a G-CW - subcomplex of EVCYC(G). Namely, we get from the Farrell-Jones Conjecture 7.2 and Theorem 8.7 an isomorphism M ~= HGn(EFIN (G); K) HGn(EVCYC(G), EFIN (G); K) -! Kn(RG). The analogous statement holds for L<-1>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 {(Mi) | i 2 I} be the set of conjugacy classes of maximal finite subgroups of G. Then (i)There is a long exact sequence M . .!.Hp+1(G\EFIN (G); K(R)) ! Kp(R[Mi]) M M i2I M ! Kp(R) HGp(EFIN (G); KR ) ! Hp(G\EFIN (G); K(R)) ! . . . i2I and analogously for L<-1>R. (ii)For R = Z there are isomorphisms M M ~= Wh n(Mi) HGn(EFIN (G), EVCYC(G); KZ) -! Wh n(G). i2I Remark 8.9. These results about groups satisfying conditions (M) and (NM) are extended in [49] to groups which map surjectively to groups satisfying cond* *i- tions (M) and (NM) with special focus on the semi-direct product of the discrete three-dimensional Heisenberg group with Z=4. 48 Remark 8.10. In [66] a special model for E_SL3(Z) is presented which allows to compute the integral group homology. Information about the algebraic K- theory of SL3(Z) can be found in [68, Chapter 7], [74]. The analysis of the other term HGn(EVCYC(G), EFIN (G); K) simplifies con- siderably under certain assumptions on G. Theorem 8.11 (On the structure of EVCYC(G)). Suppose that G satisfies the following conditions: o Every infinite cyclic subgroup C G has finite index in its centralizer CG C; o 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 [ Vmax = {NG C | C V infinite cyclic normal}. Then (a)Vmax is an infinite virtually cyclic subgroup of G and contains V ; (b)If V W G are infinite virtually cyclic subgroups of G, then Vmax = Wmax; (c)Each infinite virtually cyclic subgroup V is contained in a unique maximal infinite virtually cyclic subgroup, namely Vmax, and NG Vmax = Vmax; (ii)Let {Vi | i 2 I} be a complete system of representatives of conjugacy classes of maximal infinite virtually cyclic subgroups. Then there exists a G-pushout ` i2IG xViEFIN?(Vi) ----! EFIN?(G) pr?y ?y ` i2IG=Vi ----! EVCYC(G) whose upper horizontal arrow is an inclusion of G-CW -complexes. (iii)There are natural isomorphisms M ~= HVin(EVCYC(Vi), EFIN (Vi); KR-)!HGn(EVCYC(G), EFIN (G); KR ) i2I M <-1>~= <-1> HVin(EVCYC(Vi), EFIN (Vi); LR -)! HGn(EVCYC(G), EFIN (G); LR ). i2I 49 Proof. Each word-hyperbolic group G satisfies these two conditions by [13, The- orem 3.2 in III. .3 on page 459 and Corollary 3.10 in III. .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 2 C} in its centralizer CG dC. Let D dC be any non-trivial subgroup. Obviously dC CG D. We want to show [CG D : dC] b . e2. (8.12) Since D is central in CG D and CG D is virtually cyclic and hence |CG D=D| < 1, the spectral sequence associated to the extension 1 ! D ! CG D ! CG D=D ! 1 implies that the map D = H1(D) ! H1(CG D) is injective and has fi- nite cokernel. In particular the quotient of H1(CG D) by its torsion subgroup H1(CG D)= torsis an infinite cyclic group. Let pCGD :CG D ! H1(CG D)= tors be the canonical epimorphism. Its kernel is a finite normal subgroup. The following diagram commutes and has exact rows 1 ----! ker(pC )----! CG C --pC--!H1(CG C)= tors----! 1 ?? ? ? y ?y ?y 1 ----! ker(pD )----! CG D --pD--!H1(CG D)= tors----! 1 where the vertical maps are induced by the inclusions CG C CG D. All vertical maps are injections with finite cokernel. Fix elements zC 2 CG C and zD 2 CG D such that pC (zC ) and pD (zD ) are generators. Choose l 2 Z such that pC (zC )* * is send to l . pD (zD ). Then there is k 2 ker(pD ) with zC = k . zlD. The order of ker(pD ) divides b by assumption. If OE: ker(pD ) ! ker(pD ) is any automorphis* *m, then OEb!= id. This implies for any element k 2 ker(pD ) that d-1Y _b!-1Y !b OEi(k) = OEi(k) = 1. i=0 i=0 Hence we get in CG D if OE is conjugation with zlD d-1Y zdC= (k . zlD)d = OEi(k) . zdlD= zdlD. i=0 Obviously zD 2 CG dC since zdC= zdlDgenerates dC. Hence zeDlies in dC and we get zeD= zdfCfor some integer f. This implies zeD= zldfDand hence that l divides e. We conclude that the cokernel of the map H1(CG C)= tors! H1(CG D)= tors is bounded by e. Hence the index [CG D : CG C] is bounded by b.e since the order of ker(pD ) divides b. Since dC CG C CG dC CG D holds, equation 8.12 follows. Next we show that there is a normal infinite cyclic subgroup C0 V such that Vmax = NG C0 holds. If C0 and C00are infinite cyclic normal subgroups of 50 V , then both CG C0 and CG C00are contained in CG (C0\ C00) and C0\ C00is again an infinite cyclic normal subgroup. Hence there is a sequence of normal infinite cyclic subgroups of V dC C1 C2 C3 . . . which yields a sequence CG dC CG C1 CG C2 . .s.atisfying [ [ {CG Cn | n 1} = {CG C | C V infinite cyclic normal}. Because of 8.12 there is an upper bound on [CG Cn : CG dC] which is independent of n. Hence there is an index n0 with [ CG Cn0 = {CG C | C V infinite cyclic normal}. For any infinite cyclic subgroup C G the index of CG C in NG C is 1 or 2. Hence there is an index n1 with [ NG Cn1 = {NG C | C V infinite cyclic normal}. Thus we have shown the existence of a normal infinite cyclic subgroup C V with Vmax = 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 ` j i2IG xViEFIN?(Vi) ----! EFIN?(G) pr?y ?y ` i2IG=Vi ----! 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 XH is contractible. Given a Vi-space Y and a subgroup H G, there is after a choice of a map of sets s: G=Vi ! G, whose composition with the projection G ! G=Vi is the identity, a G-homeomorphism a -1 ~= H Y s(w) Hs(w)-! (G xViY ) , (8.13) w2G=Vi s(w)-1Hs(w) Vi which sends y 2 Y s(w)-1Hs(w)to (s(w), y). If H is infinite, the H-fixed point set of the upper right and upper left co* *rner is empty and of the lower left corner is the one-point space because of asserti* *on 51 (i)c and equation 8.13. Hence XH 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 vert* *ical arrow induces a homotopy equivalence on the H-fixed point sets. Hence XH is contractible for finite H G. This shows that X is a model for EVCYC(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 [38] 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 HVin(EVCYC(Vi), EFIN (Vi); KR ) and HVin(EVCYC(Vi), EFIN (Vi); L<-1>R) occur. They also appear in the direct sum decomposition M Kn(RVi) ~= HVin(EFIN (Vi); KR ) HVin(EVCYC(Vi), EFIN (Vi); KR ); M <-1> Ln(RVi) ~= HVin(EFIN (Vi); L<-1>R) HVin(EVCYC(Vi), EFIN (Vi); LR ). 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 Vi. They vanish for L-theory after inverting 2 by results of [19]. 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Appl., 13(2):129-165, 2000. 57 Notation cd(G), 29 cd(M), 29 CX, 33 C*r(G), 43 EF (G), 8 E_G, 8 Fn, 24 FPn, 29 FP1 , 29 G0, 15 Gd,_ 32 G , 15 JF (G), 9 J_G, 9 l(H), 33 NG H, 33 Or (G), 30 Or F(G), 30 Out (Fn), 24 P C0(G), 17 Pd(G, S), 23 T = T (G, X, X0), 20 vcd(G), 37 WG H, 33 [X, Y ]G , 6 VF , 39 Z_, 31 ß = ß(G, X, X0), 20 sg,r, 25 ALL, 8 COM, 8 COMOP, 8 FIN , 8 Tgs,r, 25 TR, 8 VCYC, 8 58 Index apartments, 22 graph of groups, 20 fundamental group, 20 building group affine, 22 almost connected, 15 Euclidean, 22 hyperbolic, 24 poly-Z, 38 cell totally disconnected, 14 equivariant closed n-dimensional virtually poly-cyclic, 38 cell, 5 equivariant open n-dimensional Hirsch rank, 38 cell, 5 hyperbolic cellular group, 24 map, 7 metric space, 24 classifying G-CW -complex for a fam- ily of subgroups, 8 length of a subgroups, 33 classifying numerable G-space for a family of subgroups, 9 mapping class group, 25 cocompact, 6 marked metric graph, 24 cohomological dimension metric space for groups, 29 hyperbolic, 24 for modules, 29 colimit topology, 5 numerable G-principal bundle, 10 component group, 15 universal, 10 component of the identity, 15 Conjecture orbit category, 30 Baum-Connes Conjecture, 43 outer space, 24 Farrell-Jones Conjecture, 43 spine of, 24 ffi-hyperbolic proper G-space, 5 group, 23 metric space, 23 resolution finite, 29 equivariant smooth triangulation, 6 free, 29 resolution, 29 F-numerable G-space, 9 Rips complex, 23 family of subgroups, 8 fundamental group of a graph of groups,skeleton, 5 20 symmetric set of generators, 23 G-CW -complex, 5 Teichmüller space, 25 finite, 6 Theorem finite dimensional, 6 A criterion for 1-dimensional mod- of dimension n, 6 els, 36 of finite type, 6 Actions on CAT(0)-spaces, 18 59 Actions on simply connected non- tree, 18 positively curved manifolds, type F Pn 18 for groups, 29 Actions on trees, 18 for modules, 29 Affine buildings, 23 type F P1 Algebraic and geometric finite- for groups, 29 ness conditions, 31 for modules, 29 Almost connected groups, 17 type VF, 39 Comparison of EF (G) and JF (G), 15 universal G-CW -complex for proper Completion Theorem for discrete G-actions, 8 groups, 44 universal G-space for proper G-actions, Discrete subgroups of almost con- 9 nected Lie groups, 18 universal numerable G-space for proper Discrete subgroups of Lie groups, G-actions, 9 37 Equivariant Cellular Approxima- virtual cohomological dimension, 36, tion Theorem, 7 37 Finiteness conditions for BG, 29 weak homotopy equivalence, 6 Homotopy characterization of EF (G),weakly contractible, 7 8 word metric, 23 Homotopy characterization of JF (G),ZOr F(G)-module, 30 9 free, 30 Models based on actions on trees, 20 On the structure of EVCYC(G), 49 Operator theoretic model, 17 Passage from topological groups to discrete groups, 32 Passage from topological groups to totally disconnected groups, 32 Passage from totally disconnected groups to discrete groups, 32 Rips complex, 23 The homotopy type of G\E_G, 41 Virtual cohomological dimension and dim(E_G, 37 Virtual cohomological dimension and the condition B(d), 36 Whitehead Theorem for Fami- lies, 6 60