Extensive amenability and an application to interval exchanges

TL;DR

This paper explores extensive amenability in group actions, demonstrating its preservation under certain constructions, and applies these findings to prove the amenability of specific subgroups of interval exchange transformations, with implications for probabilistic properties.

Contribution

It introduces a general construction preserving extensive amenability and applies it to establish amenability of subgroups of IET with rational rank ≤ 2, also providing probabilistic insights.

Findings

Extensive amenability is preserved under semidirect products.

Subgroups of IET with rational rank ≤ 2 are amenable.

Recurrent actions are shown to be extensively amenable through probabilistic methods.

Abstract

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 IET of interval exchange transformations that have angular components of rational rank~${\leq 2}$. 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 IET and show that there are subgroups $G <IET$ admitting no finitely supported measure with trivial boundary.