The growth rates of automaton groups generated by reset automata

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.