Examples of non-amenable, boundary-amenable dynamical systems

It is proved that for every $r,s\in \mathbb{N}\mathrm{\backslash }\{0,1\}$ the action of ${\mathbb{F}}_{r+s}$ on ${\mathrm{\partial }}_{\beta }({\mathbb{F}}_{r+s}\mathrm{/}{\mathbb{F}}_r)$ is topologically amenable. In particular the $C^{\ast }$-algebra associated to the corresponding quasi-regular representation has a unique non-trivial ideal. The techniques involved rely on a study of dynamical properties for actions on non-standard boundaries studied by the author and F. R{\u a}dulescu in previous works.