Examples of non-amenable, boundary-amenable dynamical systems

Abstract

Let $Γ$ be a discrete countable group with the (AP)-property. It is shown that if $Γ$ 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 $Γ$ on $\partial_β\mathfrak{X}$ is topologically amenable. The range of applications include the action of $Γ$ on $\partial_β(Γ/ Λ)$ for: (i) $Γ$ countable hyperbolic torsion-free and $Λ$ quasi-isometrically embedded with infinite index, (ii) $Γ= Λ* Λ'$ with $Λ$ non-amenable countable, $Λ'$ infinite countable and $Γ$ 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.