Lamplighter groups, bireversible automata and rational series over finite rings

Abstract

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 $2$.

23 pages, 2 figures, 4 tables, v2: updated to include full title in posting, v3: incorporates referee suggestions, final version to appear in Groups, Geometry, and Dynamics