[Paper Review] Survey on Classifying Spaces for Families of Subgroups

Wolfgang Lueck|ArXiv.org|Dec 19, 2003

Advanced Operator Algebra Research4 citations

TL;DR

This paper provides a comprehensive survey on classifying spaces for families of subgroups in topological groups, introducing two versions—G-CW-complex and numerable G-space—showing their equivalence under specific conditions. It establishes geometric models for key groups like almost connected, word hyperbolic, and arithmetic groups, and demonstrates their critical role in the Baum-Connes and Farrell-Jones conjectures by reducing K- and L-theory computations to equivariant homology on these spaces.

ABSTRACT

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

Motivation & Objective

• To define and compare two versions of classifying spaces for families of subgroups: the G-CW-complex version and the numerable G-space version.
• To construct geometric models for these classifying spaces in key cases such as almost connected groups, word hyperbolic groups, and arithmetic groups.
• To investigate finiteness conditions—finite, finite-type, and finite-dimensional models—for these classifying spaces.
• To establish the relevance of these spaces in major conjectures in K-theory and L-theory, particularly the Baum-Connes and Farrell-Jones Conjectures.
• To clarify the role of these spaces in equivariant topology, including completion theorems and classifying spaces for equivariant bundles.

Proposed method

• Define the G-CW-complex version $E_{rak{F}}(G)$ as a $G$-CW-complex with isotropy groups in the family $\frak{F}$, characterized by a universal property for $G$-equivariant maps.
• Introduce the numerable $G$-space version $J_{rak{F}}(G)$ as a $G$-space with a $G$-invariant numerable $G$-covering, also universal for $G$-equivariant maps.
• Prove that $E_{rak{F}}(G)$ and $J_{rak{F}}(G)$ are $G$-homotopy equivalent when $G$ is discrete, a Lie group with compact subgroups in $\frak{F}$, or $\frak{F}$ is the family of compact subgroups.
• Construct geometric models using symmetric spaces, Teichmüller spaces, outer space, Rips complexes, buildings, and trees for specific families and groups.
• Use equivariant homology and the induction structure to reduce computations of $K$- and $L$-groups to homology on $E_{\mathcal{F}}(G)$, particularly for the families of finite and virtually cyclic subgroups.
• Apply excision and pushout constructions to build models for $E_{\mathcal{VCYC}}(G)$ from models of $E_{\mathcal{FIN}}(V_i)$ for virtually cyclic subgroups $V_i \subseteq G$.

Experimental results

Research questions

• RQ1 Under what conditions are the $G$-CW-complex and numerable $G$-space versions of the classifying space for a family of subgroups $\frak{F}$ $G$-homotopy equivalent?
• RQ2 For which classes of topological groups (e.g., almost connected, word hyperbolic, arithmetic) do finite or finite-dimensional models exist for $E_{\mathcal{FIN}}(G)$ and $E_{\mathcal{VCYC}}(G)$?
• RQ3 How do the classifying spaces for families of subgroups facilitate the computation of algebraic $K$- and $L$-groups of group rings via the Farrell-Jones Conjecture?
• RQ4 What is the role of these classifying spaces in the formulation and proof of the Baum-Connes Conjecture for topological $K$-theory of group $C^*$-algebras?
• RQ5 How do the fixed-point sets $X^H$ for $H$ virtually cyclic or finite subgroups determine the structure of a model for $E_{\mathcal{VCYC}}(G)$?

Key findings

• The $G$-CW-complex version $E_{\frak{F}}(G)$ and the numerable $G$-space version $J_{\frak{F}}(G)$ are $G$-homotopy equivalent when $G$ is discrete, a Lie group with compact subgroups in $\frak{F}$, or $\frak{F}$ is the family of compact subgroups.
• For discrete groups with torsion, $E_{\mathcal{FIN}}(G)$ can admit a finite-dimensional model even when $EG$ cannot, with the minimal dimension related to the virtual cohomological dimension.
• A $G$-pushout construction using models of $E_{\mathcal{FIN}}(V_i)$ for virtually cyclic subgroups $V_i \subseteq G$ yields a model for $E_{\mathcal{VCYC}}(G)$, with $X^H$ contractible for all finite and infinite virtually cyclic $H \subseteq G$.
• The spaces $E_{\mathcal{FIN}}(G)$ and $E_{\mathcal{VCYC}}(G)$ provide geometric models for the relevant equivariant homology theories in the Farrell-Jones Conjecture, reducing $K_n(RG)$ and $L_n(RG)$ to equivariant homology groups.
• The terms $H_n^{V_i}(E_{\mathcal{VCYC}}(V_i), E_{\mathcal{FIN}}(V_i); \mathbf{K}_R)$ and $H_n^{V_i}(E_{\mathcal{VCYC}}(V_i), E_{\mathcal{FIN}}(V_i); \mathbf{L}_R^{\langle -\infty \rangle})$ contain Nil and UNIL-terms in algebraic $K$- and $L$-theory of $V_i$, which vanish rationally for $R = \mathbb{Z}$ and after inverting 2 in $L$-theory.
• For any $CW$-complex $X$, there exists a discrete group $G$ such that the orbit space $G \backslash E_{\mathcal{FIN}}(G)$ is homotopy equivalent to $X$, showing the universality of these orbit spaces in homotopy theory.