Selfless inclusions arising from commensurator groups of hyperbolic groups

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