The growth rates of automaton groups generated by reset automata

TL;DR

This paper establishes conditions under which automaton groups generated by reset automata have infinite order and contain free semigroups, providing new insights into their growth and structure.

Contribution

It introduces sufficient conditions for infinite order in automaton groups within a class including reset automata, and offers a new proof related to group growth and free semigroups.

Findings

Groups generated by automata in class C are infinite under certain conditions.

Infinite automaton groups contain free semigroups of rank at least 2.

Provides a new proof of a result by Chou on group growth and free semigroups.

Abstract

We give sufficient conditions for when groups generated by automata in a class $\mathcal{C}$ of transducers, which contains the class of reset automata transducers, have infinite order. As a consequence we also demonstrate that if a group generated by an automata in $\mathcal{C}$ is infinite, then it contains a free semigroup of rank at least 2. This gives a new proof, in the context of groups generated by automaton in $\mathcal{C}$, of a result of Chou showing that finitely generated elementary amenable groups either have polynomial growth or contain a free semigroup of rank at least 2.