Lamplighter groups, bireversible automata and rational series over finite rings

TL;DR

This paper constructs lamplighter groups as automaton groups using affine transformations over power series rings with finite ring coefficients, and characterizes when these automata are bireversible based on the structure of the abelian group.

Contribution

It introduces a new method to realize lamplighter groups as automaton groups via affine transformations over finite rings and provides a criterion for bireversibility.

Findings

Lamplighter groups realized as automaton groups via affine transformations.

Bireversibility depends on the structure of the 2-Sylow subgroup of the abelian group.

Characterization of when automaton groups are bireversible based on group decomposition.

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 $2$.