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

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

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