Self-similar groups and the zig-zag and replacement products of graphs

Abstract

Every finitely generated self-similar group naturally produces an infinite sequence of finite $d$-regular graphs ${\Gamma }_n$. We construct self-similar groups, whose graphs ${\Gamma }_n$ can be represented as an iterated zig-zag product and graph powering: $Γ_{n+1}=Γ_n^k\mathop{\mbox{\textcircled{$z$}}}Γ$ ($k\ge 1$). Also we construct self-similar groups, whose graphs ${\Gamma }_n$ can be represented as an iterated replacement product and graph powering: $Γ_{n+1}=Γ_n^k\mathop{\mbox{\textcircled{$r$}}}Γ$ ($k\ge 1$). This gives simple explicit examples of self-similar groups, whose graphs ${\Gamma }_n$ form an expanding family, and examples of automaton groups, whose graphs ${\Gamma }_n$ have linear diameters $\mathrm{diam}({\Gamma }_n)=O(n)$ and bounded girth.

11 pages

References in corpus (1)

• Lifts, derandomization, and diameters of Schreier graphs of Mealy automata