On the classifying space for proper actions of groups with cyclic torsion

TL;DR

This paper develops a unified topological framework for the Baum-Connes conjecture applicable to various groups with cyclic torsion, computing their Bredon homology using specialized models of universal spaces for proper actions.

Contribution

It introduces a general approach to analyze the topological aspects of the Baum-Connes conjecture for groups with cyclic torsion, extending existing methods to new classes of groups.

Findings

Computed Bredon homology for groups with aspherical presentations

Extended models of universal spaces for proper actions to new group classes

Provided insights into the torsion structure of specific groups

Abstract

In this paper we introduce a common framework for describing the topological part of the Baum-Connes conjecture for a wide class of groups. We compute the Bredon homology for groups with aspherical presentation, one-relator quotients of products of locally indicable groups, extensions of $\mathbb{Z}^n$ by cyclic groups, and fuchsian groups. We take advantage of the torsion structure of these groups to use appropriate models of the universal space for proper actions which allow us, in turn, to extend some technology defined by Mislin in the case of one-relator groups.