Title: Survey on Classifying Spaces for Families of Subgroups
Author: Wolfgang Lueck
AMS Classification numbers: 55R35, 57S99, 20F65, 18G99
Address: Mathematisches Institut der
Westfaelischen Wilhelms Universitaet
Einsteinstr. 62
48149 Muenster
Germany
Abstract:
We define for a topological group G and a family of subgroups
F two versions for the classifying space for the family F,
the G-CW-version E_F(G) and the numerable G-space
version J_F(G). They agree if G is discrete, or if G is a
Lie group and each element in F compact, or if 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.