Amenability of linear-activity automaton groups

TL;DR

This paper proves that all linear-activity automaton groups are amenable by analyzing the entropy of specific random walks, resolving open questions in the field.

Contribution

It establishes the amenability of linear-activity automaton groups, answering open problems posed by Nekrashevich and Sidki.

Findings

All linear-activity automaton groups are amenable.

A symmetric random walk on the mother group has asymptotic entropy zero.

The proof involves constructing a degree 1 automaton group with specific properties.

Abstract

We prove that every linear-activity automaton group is amenable. The proof is based on showing that a sufficiently symmetric random walk on a specially constructed degree 1 automaton group -- the mother group -- has asymptotic entropy 0. Our result answers an open question by Nekrashevich in the Kourovka notebook, and gives a partial answer to a question of Sidki.