Coarse geometry of Hecke pairs and the Baum-Connes conjecture

TL;DR

This paper investigates the coarse geometric properties of Hecke pairs and their impact on the Baum-Connes conjecture, establishing new stability results and expanding the class of groups known to satisfy the conjecture with coefficients.

Contribution

It introduces new stability results for the Baum-Connes and Novikov conjectures for co-Haagerup Hecke pairs, generalizing previous work and providing new examples of groups satisfying the conjecture.

Findings

Proves stability results for Baum-Connes conjecture in co-Haagerup cases

Shows certain S-arithmetic subgroups of Sp(5,1) and Sp(3,1) satisfy the conjecture with coefficients

Extends known classes of groups for which the Baum-Connes conjecture holds

Abstract

We study Hecke pairs using the coarse geometry of their coset space and their Schlichting completion. We prove new stability results for the Baum-Connes and the Novikov conjectures in the case where the pair is co-Haagerup. This allows to generalize previous results, while providing new examples of groups satisfying the Baum-Connes conjecture with coefficients. For instance, we show that for some S-arithmetic subgroups of Sp(5,1) and Sp(3,1) the conjecture with coefficients holds.