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.

The Lamplighter Group

“…This section concerns with f (x) = 1 1 − X only. The above theorem tells us [4] that we can associate to µ[f ] a Mealy machine whose state functions are given by the states of µ[f ]. In other words, define…”

“…

We are following [4]. Nevertheless we are interested only in claryfication that the lamplighter group can be realized as a 2-states Mealy machine.

…”

Bireversible automata generating lamplighter groups

“…As in the work of Skipper and Steinberg [11], we will make use of formal power series to understand our groups. Our main innovation here is that, instead of equipping finite abelian groups with a ring structure, we will use the fact that every abelian group comes equipped with a natural module structure over the ring of its endomorphisms, which gives us more flexibility.…”

“…Later, Bondarenko, D'Angeli and Rodaro constructed a bireversible automaton generating the group $$(\mathbb {Z}/3\mathbb {Z})\wr \mathbb {Z}$$ [1], thus obtaining the first (infinite) solvable example. This led to an interest in understanding which groups of the form $$A\wr \mathbb {Z}$$ with $$A$$ finite abelian, sometimes known as lamplighter groups , can be obtained from such automata, culminating in the recent work of Skipper and Steinberg [11], who proved that if $$A$$ satisfies some technical condition on the Sylow 2‐subgroup of the group $$A$$ (we refer the reader to [11] for the precise condition), then $$A\wr \mathbb {Z}$$ can be generated by a bireversible automaton. Unfortunately, their result does not say anything about the groups not satisfying this condition, and in particular, it does not cover the case of the classical lamplighter group $$(\mathbb {Z}/2\mathbb {Z})\wr \mathbb {Z}$$.…”

“…A bireversible automaton whose group is a semidirect product of a lamplighter group with the cyclic group of order 2 is presented in [18]. The newest examples of bireversible automata whose groups are lamplighter groups were constructed in [1], [20] and [7], where detailed overview of already mentioned and other related results can be found.…”