Abelian Invertible Automata

Invertible transducers are particular Mealy automata that define so-called automata groups, subgroups of the full automorphism group of the infinite binary tree. In the recent past, automata groups have become a major source of interesting and challenging constructions in group theory. While this research typically focuses on properties of the associated groups, we describe the topological structure of the associated automata in the special case where the group in question is free Abelian. As it turns out, there are connections between these automata and the theory of algebraic number fields as well as the theory of tiles. We conclude with a conjecture about the connectivity properties of the canonical invertible automata generating free Abelian groups.