Survey on Classifying Spaces for Families of Subgroups

By W. Lück. 2003

270 citations recorded by the source corpus.

We define for a topological group G and a family of subgroups \( \mathcal{F} \) two versions for the classifying space for the family \( \mathcal{F} \) , the G-CW-version \( E_\mathcal{F} \) (G) and the numerable G-space version \( J_\mathcal{F} \) (G). They agree if G is discrete, or if G is a Lie group and each element in \( \mathcal{F} \) compact, or if \( \mathcal{F} \) is the family of compact subgroups. We discuss special geometric models for these spaces for the family of compact open groups in special cases such as almost connected groups G and word hyperbolic groups G. We deal with the question whether there are finite models, models of finite type, finite dimensional models. We also discuss the relevance of these spaces for the Baum-Connes Conjecture about the topological K-theory of the reduced group C*-algebra, for the Farrell-Jones Conjecture about the algebraic K- and L-theory of group rings, for Completion Theorems and for classifying spaces for equivariant vector bundles and for other situations.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9e0b421b466b95897808ef9284298366e7915be5

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

The K-theoretic Farrell–Jones conjecture for hyperbolic groups (2007, background): “We denote by EF G the associated classifying space of the family F (see for instance [45]).”
Cohomological finiteness conditions for elementary amenable groups (2009, background): “If G is polycyclic-by-finite it is of type Bredon F, which can be seen by an induction on the Hirsch length of the group, see [26], Example 5.”
On the classifying space of the family of virtually cyclic subgroups (2007, background): “The same question for EG has already been thoroughly investigated, and in many cases satisfying answers are known (see for instance [16], where further references to relevant papers are given).”
The Farrell-Jones Conjecture for cocompact lattices in virtually connected Lie groups (2011, background): “JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 27, Number 2, April 2014, Pages 339–388 S 0894-0347(2014)00782-7 Article electronically published on January 15, 2014 THE FARRELL-JONES CONJECTURE FOR COCOMPACT LATTICES IN VIRTUALLY CONNECTED LIE GROUPS”
Isomorphism Conjecture for homotopy K-theory and groups acting on trees (2004, background): “For more information about the spaces we refer for instance to [17].”