A localization algebra approach to the Baum-Connes conjecture for extensions

TL;DR

This paper uses a localization algebra approach to prove that the Baum-Connes conjecture with coefficients for a group extension can be deduced from the conjecture holding for the quotient and preimages of finite subgroups.

Contribution

It introduces a novel localization algebra method to establish the conjecture's validity for group extensions based on known cases.

Findings

Proves the Baum-Connes conjecture for extensions using localization algebra techniques.

Shows the conjecture holds for extensions if it holds for the quotient and finite subgroup preimages.

Provides a new proof framework for the conjecture in the context of group extensions.

Abstract

For an extension $1\rightarrow N \rightarrow \Gamma \xrightarrow{q} \Gamma / N \rightarrow 1$ of discrete countable groups, it is known that the Baum-Connes conjecture with coefficients holds for $\Gamma$ if it holds for $\Gamma / N$ and $q^{-1}(F)$ for any finite subgroup $F$ of $\Gamma / N$. In this paper, we employ a localization algebra approach to prove this result.