Automaton (semi)groups: Wang tilings and Schreier tries

Groups and semigroups generated by Mealy automata were formally introduced in the early sixties. They revealed their full potential over the years, by contributing to important conjectures in group theory. In the current chapter, we intend to provide various combinatorial and dynamical tools to tackle some decision problems all related to some extent to the growth of automaton (semi)groups. In the first part, we consider Wang tilings as a major tool in order to study and understand the behavior of automaton (semi)groups. There are various ways to associate a Wang tileset with a given complete and deterministic Mealy automaton, and various ways to interpret the induced Wang tilings. We describe some of these fruitful combinations, as well as some promising research opportunities. In the second part, we detail some toggle switch between a classical notion from group theory—Schreier graphs—and some properties of an automaton group about its growth or the growth of its monogenic subgroups. We focus on polynomial-activity automata and on reversible automata, which are somehow diametrically opposed families.