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$C^*$-irreducibility of commensurated subgroups

Kang Li and Eduardo Scarparo

Vol. 322 (2023), No. 2, 369–380
Abstract

Given a commensurated subgroup Λ of a group Γ, we completely characterize when the inclusion Λ Γ is C-irreducible and provide new examples of such inclusions. In particular, we obtain that PSL (n, ) PGL (n, ) is C-irreducible for any n , and that the inclusion of a C-simple group into its abstract commensurator is C-irreducible.

The main ingredient that we use is the fact that the action of a commensurated subgroup Λ Γ on its Furstenberg boundary FΛ can be extended in a unique way to an action of Γ on FΛ. 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
Authors
Kang Li
Friedrich-Alexander-Universität Erlangen-Nürnberg
Erlangen
Germany
Eduardo Scarparo
Center for Engineering
Federal University of Pelotas
Pelotas
Brazil

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