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 * \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–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.