Direct Splitting Method for the Baum-Connes Conjecture

TL;DR

This paper introduces the direct splitting method, a new approach to analyze the Baum-Connes conjecture, simplifying proofs for certain cases and paving the way for finite-dimensional proofs in specific group actions.

Contribution

The paper presents the direct splitting method, a novel technique that simplifies existing proofs of the Baum-Connes conjecture and facilitates finite-dimensional approaches for particular group actions.

Findings

Simplifies proofs of known Baum-Connes cases

Provides a new framework for finite-dimensional proofs

Enhances understanding of group actions on CAT(0)-cubical spaces

Abstract

We introduce a new method for studying the Baum-Connes conjecture, which we call the direct splitting method. The method can simplify and clarify proofs of some of the known cases of the conjecture. In a separate paper, with J. Brodzki, E. Guentner and N. Higson, a similar idea will be used to give a finite-dimensional proof of the Baum-Connes conjecture for groups which act properly and co-compactly on a finite-dimensional CAT(0)-cubical space.