A connected 3-state reversible Mealy automaton cannot generate an infinite Burnside group

TL;DR

This paper proves that connected 3-state invertible-reversible Mealy automata cannot generate infinite Burnside groups, extending previous results and introducing new techniques for analyzing automaton groups.

Contribution

It establishes a new limitation on automaton groups, showing that connected 3-state invertible-reversible Mealy automata cannot produce infinite Burnside groups, using novel methods.

Findings

Connected 3-state invertible-reversible Mealy automata cannot generate infinite Burnside groups.

Introduces new techniques for constructing elements of infinite order in automaton groups.

Extends previous results from 2-state to 3-state automata.

Abstract

The class of automaton groups is a rich source of the simplest examples of infinite Burnside groups. However, there are some classes of automata that do not contain such examples. For instance, all infinite Burnside automaton groups in the literature are generated by non reversible Mealy automata and it was recently shown that 2-state invertible-reversible Mealy automata cannot generate infinite Burnside groups. Here we extend this result to connected 3-state invertible-reversible Mealy automata, using new original techniques. The results provide the first uniform method to construct elements of infinite order in each infinite group in this class.