$C^*$-irreducibility of commensurated subgroups

Kang Li and Eduardo Scarparo

Vol. 322 (2023), No. 2, 369–380

DOI: 10.2140/pjm.2023.322.369

Abstract

Given a commensurated subgroup $\Lambda$ of a group $\Gamma$, we completely characterize when the inclusion $\Lambda \le \Gamma$ is $C^{\ast }$-irreducible and provide new examples of such inclusions. In particular, we obtain that $\mathrm{PSL}<!--FUNCTION\; APPLICATION-->(n,\mathbb{Z})\le \mathrm{PGL}<!--FUNCTION\; APPLICATION-->(n,\mathbb{Q})$ is $C^{\ast }$-irreducible for any $n\in \mathbb{N}$, and that the inclusion of a $C^{\ast }$-simple group into its abstract commensurator is $C^{\ast }$-irreducible.

The main ingredient that we use is the fact that the action of a commensurated subgroup $\Lambda \le \Gamma$ on its Furstenberg boundary ${\partial }_F\Lambda$ can be extended in a unique way to an action of $\Gamma$ on ${\partial }_F\Lambda$. Finally, we also investigate the counterpart of this extension result for the universal minimal proximal space of a group.

Keywords

commensurated subgroups, C*-simplicity, Furstenberg boundary

Mathematical Subject Classification

Primary: 22D25, 37B05

Milestones

Received: 30 December 2022

Accepted: 18 February 2023

Published: 23 May 2023