The Liouville property for groups acting on rooted trees

TL;DR

This paper proves that groups generated by bounded activity automata on rooted trees have the Liouville property for symmetric, finitely supported measures, and provides entropy bounds for such measures in automaton groups.

Contribution

It establishes the Liouville property for a broad class of automorphism groups of rooted trees and offers uniform entropy bounds for automaton groups.

Findings

Groups generated by bounded activity automata have the Liouville property.

The Liouville property extends to all automorphisms of bounded type of a rooted tree.

A uniform upper bound for entropy of convolutions in automaton groups is provided.

Abstract

We show that on groups generated by bounded activity automata, every symmetric, finitely supported probability measure has the Liouville property. More generally we show this for every group of automorphisms of bounded type of a rooted tree. For automaton groups, we also give a uniform upper bound for the entropy of convolutions of every symmetric, finitely supported measure.