Selfless inclusions arising from commensurator groups of hyperbolic groups

TL;DR

This paper introduces new examples of C*-selfless groups by analyzing the commensurator groups of hyperbolic groups, showing they have topologically free boundaries.

Contribution

It proves that the commensurator group of a torsion-free hyperbolic group is C*-selfless, expanding understanding of group inclusions and boundaries.

Findings

Comm(H) is C*-selfless for torsion-free hyperbolic H

The Gromov boundary is a topologically free extreme boundary

Applicable to groups containing H in an almost normal way

Abstract

We provide new examples of $\mathrm{C}^*$-selfless groups and inclusions. In particular, we prove that the commensurator group ${\rm Comm}(H)$ of a torsion-free hyperbolic group $H$ is $\mathrm{C}^*$-selfless. Our approach involves showing that the Gromov boundary $\partial H$ is a topologically free extreme boundary for ${\rm Comm}(H)$, ${\rm Aut}(H)$, and for other groups that contain $H$ in an almost normal way.