Some Solvable Automaton Groups

It is shown that certain ascending HNN extensions of free abelian groups of finite rank, as well as various lamplighter groups, can be realized as automaton groups, i.e., can be given a self-similar structure. This includes the solvable Baumslag-Solitar groups BS(1, m), for m 6= ±1. In addition, it is shown that, for any relatively prime integers m, n ≥ 2, the pair of Baumslag-Solitar groups BS(1, m) and BS(1, n) can be realized by a pair of dual automata. The examples are then used to illustrate more general connections between Schreier graphs, composition of automata and dual automata. Groups generated by automata appeared already in the 1950’s. Among the pioneering works we mention Horejs [Hoř63] and Aleshin [Ale72]. Important examples appeared later, in particular the well known examples of infinite residually finite torsion groups, and groups of intermediate growth constructed by Grigorchuk in [Gri80, Gri83]. Many groups were then shown to belong to that class; in particular linear groups over Z [BS98]. The set of all transformations generated by finite automata over a fixed finite alphabet form a group, denoted F . It is not known which solvable groups appear as subgroups of F , i.e., appear as groups generated by finite automata. Progress in this direction has been achieved in the works of Sidki and Brunner [BS02, Sid03, Sid]. In this note, we are interested in (solvable) groups that are generated by all the states of a single finite automaton. Such groups are called automaton groups. The special interest in this more restricted setting is justified by the self-similarity structure that is apparent as soon as a group is realized as an automaton group. The purpose of this note is twofold. We go over some well known notions and constructions (automaton groups, inversion, composition) as well as some less known (dual automata). At the same time, we realize some solvable groups as automaton groups (thus giving them self-similar structure) and use them to illustrate the introduced notions. For example, we show that, for any n coprime to m, the solvable BaumslagSolitar groups BS(1,m) = 〈 a, t | tat = a 〉 belong to the class of automaton groups on a n-letter alphabet. The automata that describe them are related to multiplication by m and addition in base n. Similar constructions, corresponding to multiplication by linear polynomials over the finite ring Z/nZ, lead to “lamplighter groups”, i.e. the groups Ln = (Z/nZ) ≀ Z = 〈 a, t | a n = [a, tat] = 1 ∀i ∈ Z 〉. Date: March 14, 2006. 1991 Mathematics Subject Classification. 20E08, 68Q70, 20F05.