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

TL;DR

This paper constructs self-similar groups generating infinite sequences of finite regular graphs using zig-zag and replacement products, providing explicit examples of expanding families with linear diameters and bounded girth.

Contribution

It introduces new self-similar groups whose graphs are formed via iterated zig-zag and replacement products, expanding the class of known automaton groups with specific graph properties.

Findings

Graphs form expanding families.

Graphs have linear diameters, O(n).

Graphs exhibit bounded girth.

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.