To Infinity and Beyond

We prove that if a group generated by a bireversible Mealy automaton contains an element of infinite order, its growth blows up and is necessarily exponential. As a direct consequence, no infinite virtually nilpotent group can be generated by a bireversible Mealy automaton. The study on how (semi)groups grow has been highlighted since Milnor’s question on the existence of groups of intermediate growth (faster than any polynomial and slower than any exponential) in 1968 [12], and the very first example of such a group given by Grigorchuk [5]. Uncountably many examples have followed this first one, see for instance [6]. Bartholdi and Erschler have even obtained results on precise computations of growth, in particular they proved that if a function satisfies some frame property, then there exists a finitely generated group with growth equivalent to it [1]. Besides, for now, intermediate growth and automaton groups, that is groups generated by Mealy automata, seem to have a very strong link, since the only known examples of intermediate growth groups are either automaton groups, or based on such groups. There exists no criterium to test if a Mealy automaton generates a group of intermediate growth and it is not even known if this property is decidable. However, there is no known example in the litterature of a bireversible Mealy automaton generating an intermediate growth group and it is legitimate to wonder if it is possible. This article enter in this scope. We prove that if there exists at least one element of infinite order in a group generated by a bireversible Mealy automaton, then its growth is necessarily exponential. It has been conjuctered, and proved in some cases [4], that an infinite group generated by a bireversible Mealy automaton always has an element of infinite order, which suggests that, indeed, a group generated by a bireversible Mealy automaton either is finite, or has exponential growth. Finally, let us mention the work by Brough and Cain to obtain some criteria to decide if a semigroup is an automaton semigroup [2]. Our work can be seen as partially answering a similar question: can a given group be generated by a bireversible Mealy automaton? A consequence of our result is that no infinite virtually nilpotent group can be. This article is organized as follows. In Section 1, we define the automaton groups and the growth of a group, and give some properties on the connected components of the powers of a Mealy automaton. In Section 2, we study the behaviour of some equivalence classes of words on the state set of a Mealy automaton. Finally, the main result takes place in Section 3.