Coarse geometry of Hecke pairs and the Baum–Connes conjecture

Clément Dell’Aiera

Vol. 322 (2023), No. 1, 21–37

DOI: 10.2140/pjm.2023.322.21

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 $\mathrm{Sp}<!--FUNCTION\; APPLICATION-->(5,1)$ and $\mathrm{Sp}<!--FUNCTION\; APPLICATION-->(3,1)$ the conjecture with coefficients holds.

Keywords

$K`$-theory and homology, algebraic topology, operator algebras

Mathematical Subject Classification

Primary: 19K35, 22-D-55, 46L80, 55-N-20

Milestones

Received: 27 June 2022

Revised: 3 January 2023

Accepted: 8 January 2023

Published: 3 May 2023