On amenability of automata groups

TL;DR

This paper proves that groups generated by bounded automata acting on rooted trees are amenable, using analysis of random walks on specific 'Mother groups' to establish the result.

Contribution

It introduces a novel approach to proving amenability for automata groups by reducing the problem to analyzing 'Mother groups' and their asymptotic properties.

Findings

Bounded automata groups are amenable.

Amenability extends to many classes of automata-generated groups.

Analysis of random walks on 'Mother groups' is key to the proof.

Abstract

We show that the group of bounded automatic automorphisms of a rooted tree is amenable, which implies amenability of numerous classes of groups generated by finite automata. The proof is based on reducing the problem to showing amenability just of a certain explicit family of groups ("Mother groups") which is done by analyzing the asymptotic properties of random walks on these groups.