Selfless inclusions arising from commensurator groups of hyperbolic groups

Abstract

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

9 pages, comments welcome!