Extensions of amenable groups by recurrent groupoids

TL;DR

This paper establishes a new criterion for the amenability of groups acting by homeomorphisms, linking local action properties and recurrency of orbital Schreier graphs, unifying many known cases and solving open problems.

Contribution

It introduces a general method to prove group amenability based on local action properties and recurrency, covering numerous previously unresolved cases.

Findings

Proves amenability for Grigorchuk's group and Basilica group

Unifies various classes of groups under a common amenability criterion

Provides a new approach to analyze groups in holomorphic dynamics

Abstract

We show that amenability of a group acting by homeomorphisms can be deduced from a certain local property of the action and recurrency of the orbital Schreier graphs. This covers amenability of a wide class groups, the amenability of which was an open problem, as well as unifies many known examples to one general proof. In particular, this includes Grigorchuk's group, Basilica group, the full topological group of Cantor minimal system, groups acting on rooted trees by bounded automorphisms, groups generated by finite automata of linear activity growth, groups that naturally appear in holomorphic dynamics.