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]. For R = Z they vanishes rationally for algebraic K-theory by results of
[44].
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[80]B. Wilking. On fundamental groups of manifolds of nonnegative curvature.
Differential Geom. 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