Examples of non-amenable, boundary-amenable dynamical systems

Abstract

Let $\Gamma$ be a discrete countable group with the (AP)-property. It is shown that if $\Gamma$ acts on a countable set $\mathfrak{X}$ in such a way that the infinite intersection of stabilizer subgroups is always trivial, then the induced action of $\Gamma$ on ${\partial }_{\beta }\mathfrak{X}$ is topologically amenable. The range of applications includes the action of $\Gamma$ on ${\partial }_{\beta }(\Gamma /\Lambda )$ for: (i) $\Gamma$ countable hyperbolic torsion-free and $\Lambda$ quasi-isometrically embedded with infinite index; (ii) $\Gamma =\Lambda \ast {\Lambda }^{\prime }$ with $\Lambda$ non-amenable countable, ${\Lambda }^{\prime }$ infinite countable and $\Gamma$ with the (AP)-property; moreover, this includes the case of actions of groups of automorphisms of a $k$-regular tree with $k\ge 3$ generated by a finite number of Haar-random elements on the Stone–Čech boundary of the tree. The techniques involved rely on a study of dynamical properties for actions on non-standard boundaries studied by the author and F. Rădulescu in previous works.