Examples of non-amenable, boundary-amenable dynamical systems

Let Γ be a discrete countable group with the (AP)-property. It is shown that if Γ acts on a countable set X in such a way that the infinite intersection of stabilizer subgroups is always trivial, then the induced action of Γ on ∂β X is topologically amenable. The range of applications includes the action of Γ on ∂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 ≥ 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.