On level-transitivity and exponential growth

We prove that if the group generated by an invertible and reversible Mealy automaton acts level-transitively on a regular rooted tree, then the semigroup generated by the dual automaton has exponential growth, hence giving a decision procedure of exponential growth for a restricted family of automaton (semi)groups. The purpose of this note is to link up two classes of groups and semigroups highly studied for themselves: level-transitive (semi)groups and (semi)groups of exponential growth, through automaton (semi)groups. On the one hand, level-transitive groups (or equivalently spherically transitive groups, depending on the authors) — i.e. groups acting transitively on every level of a regular rooted tree — have received special focus these last years because of branch groups, which form a particular class of level-transitive groups, one of the three classes into which the class of just infinite groups is naturally decomposed [8, 3]. On the other hand, the study on how (semi)groups grow has been highlighted since Milnor’s question on the existence of groups of intermediate growth in 1968 [13] and the very first example of such a group given by Grigorchuk [6]. In this note, we prove that no semigroup of polynomial or intermediate growth can be generated by an invertible and reversible Mealy automaton whose dual generates a level-transitive group. Even if the problem of deciding the leveltransitivity of an automaton group is still open, there exist some families of Mealy automata for which the level-transitivity of an element in the generated semigroup is decidable [15].