Examples of non-amenable, boundary-amenable dynamical systems

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 include 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) $\Gamma= \Lambda * \Lambda '$ with $\Lambda$ non-amenable countable, $\Lambda'$ 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 $k \geq 3$ generated by a finite number of Haar-random elements on the Stone-{\v C}ech boundary of the tree.