Extensive amenability and an application to interval exchanges

By Kate Juschenko, N. Matte Bon, Nicolas Monod, Mikael de la Salle. Ergodic Theory and Dynamical Systems, 2015

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Extensive amenability is a property of group actions which has recently been used as a tool to prove amenability of groups. We study this property and prove that it is preserved under a very general construction of semidirect products. As an application, we establish the amenability of all subgroups of the group $\text{IET}$ of interval exchange transformations that have angular components of rational rank less than or equal to two. In addition, we obtain a reformulation of extensive amenability in terms of inverted orbits and use it to present a purely probabilistic proof that recurrent actions are extensively amenable. Finally, we study the triviality of the Poisson boundary for random walks on $\text{IET}$ and show that there are subgroups $G<\text{IET}$ admitting no finitely supported measure with trivial boundary.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/29cbcfc0dc81d5055d73a8a7c63bef7c0a0d38cf