Lamplighter groups, bireversible automata, and rational series over finite rings (Q784896)

Summary: We realize lamplighter groups \(A \wr \mathbb{Z}\), with \(A\) a finite abelian group, as automaton groups via affine transformations of power series rings with coefficients in a finite commutative ring. Our methods can realize \(A \wr \mathbb{Z}\) as a bireversible automaton group if and only if the 2-Sylow subgroup of \(A\) has no multiplicity one summands in its expression as a direct sum of cyclic groups of order a power of 2.