On reversible automata generating lamplighter groups (Q6632100)

Let \(X\) be a finite abelian group. The authors define an oriented square complex \(\Delta_{X}\) such that it defines a reversible (but not bireversible) automaton \(\mathcal{A}_{X}\). Both the set of states of \(\mathcal{A}_{X}\) and its alphabet are the underlying set of \(X\). Then its transition and output functions are defined by the left regular action of \(X\).\N\NThe main result of the paper under review is the following statement. For every abelian group \(X\) of order \(n \geq 2\) the group of the automaton \(\mathcal{A}_{X}\) is isomorphic to the lamplighter group \(X \wr \mathbb{Z}\). In particular, it means that the finite state wreath power (see [\textit{A. S. Oliyny}, Semigroup Forum 82, No. 3, 423--436 (2011; Zbl 1229.20069)]) of a regular group \(X\) contains \(X \wr \mathbb{Z}\).