Examples of non-amenable, boundary-amenable dynamical systems

Abstract

Let $\hat{I}“$ be a discrete countable group with the (AP)-property. It is shown that if $\hat{I}“$ 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 $\hat{I}“$ on ${\partial }_{\hat{I}}^2\mathfrak{X}$ is topologically amenable. The range of applications include the action of $\hat{I}“$ on ${\partial }_{\hat{I}}^2(\hat{I}“/\hat{I}›)$ for: (i) $\hat{I}“$ countable hyperbolic torsion-free and $\hat{I}›$ quasi-isometrically embedded with infinite index, (ii) $\hat{I}“=\hat{I}›\ast \hat{I}{›}^{\prime }$ with $\hat{I}›$ non-amenable countable, $\hat{I}{›}^{\prime }$ infinite countable and $\hat{I}“$ 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-{\v C}ech boundary of the tree.

16 pages. This is the final version, accepted for publication in Groups Geom. Dyn. The bibliography has been updated