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: $\Gamma_{n+1}=\Gamma_n^k\mathop{\mbox{\textcircled{$z$}}}\Gamma$ ($k\geq 1$). Also we construct self-similar groups, whose graphs $\Gamma_n$ can be represented as an iterated replacement product and graph powering: $\Gamma_{n+1}=\Gamma_n^k\mathop{\mbox{\textcircled{$r$}}}\Gamma$ ($k\geq 1$). 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 ${\rm diam}(\Gamma_n)=O(n)$ and bounded girth.