# zbMATH — the first resource for mathematics

Some solvable automaton groups. (English) Zbl 1106.20021
Grigorchuk, Rostislav (ed.) et al., Topological and asymptotic aspects of group theory. AMS special session on probabilistic and asymptotic aspects of group theory, Athens, OH, USA, March 26–27, 2004 and the AMS special session on topological aspects of group theory, Nashville, TN, USA, October 16–17, 2004. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3756-7/pbk). Contemporary Mathematics 394, 11-29 (2006).
Summary: 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 (and can therefore be given a self-similar structure). This includes the solvable Baumslag-Solitar groups $$BS(1,m)$$, for $$m\neq\pm 1$$.
In addition, it is shown that, for any relatively prime integers $$m,n\geq 2$$, 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.
For the entire collection see [Zbl 1085.20501].

##### MSC:
 20E08 Groups acting on trees 20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations 20F05 Generators, relations, and presentations of groups 20F16 Solvable groups, supersolvable groups 20F65 Geometric group theory 68Q70 Algebraic theory of languages and automata