Some solvable automaton groups

Abstract

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 not +/-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.