Strong bolicity and the Baum-Connes conjecture for relatively hyperbolic groups

TL;DR

This paper constructs a strongly bolic metric for certain relatively hyperbolic groups, enabling the proof of the Baum-Connes conjecture for these groups under specific conditions, using advanced geometric and algebraic techniques.

Contribution

It introduces a new strongly bolic metric for a class of relatively hyperbolic groups and applies Lafforgue's theorem to prove the Baum-Connes conjecture in this context.

Findings

Baum-Connes conjecture proven for these groups

Construction of strongly bolic metrics for specific groups

Use of random coset representatives called masks

Abstract

We construct a strongly bolic metric for a certain class of relatively hyperbolic groups, which includes those with CAT(0) parabolics and virtually abelian parabolics. If we further assume that the parabolics satisfy (RD), applying a theorem of Lafforgue, we deduce the Baum-Connes conjecture for these groups. One of the key ingredients in our construction is the use of random coset representatives called masks, developed by Chatterji and Dahmani.