On bireversible bireversible Mealy automata and the Burnside problem

There exist undoubtedly strong links between the Burnside problem and the class of automaton groups. Indeed, many interesting examples of infinite Burnside groups are automaton groups, in particular for any prime p, there exist an infinite Burnise p-group generated by a Mealy automaton. Moreover the simplest known example of an infinite Burnside group arises in this class. However there is no known example of such a group generated by a reversible Mealy automaton. In this paper we prove that, in fact, no connected reversible Mealy automaton of prime size can generate an infinite Burnside group. In addition we explain how our method can be applied for some Mealy automata of non prime size. (C) 2017 Elsevier B.V. All rights reserved.