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 Γn. We construct self-similar groups, whose graphs Γn can be represented as an iterated zig-zag product and graph powering: Γn+1=Γnkz⃝Γ (k≥1). We also construct self-similar groups, whose graphs Γn can be represented as an iterated replacement product and graph powering: Γn+1=Γnkr⃝Γ (k≥1). This gives simple explicit examples of self-similar groups, whose graphs Γn form an expanding family, and examples of automaton groups, whose graphs Γn have linear diameters diam(Γn)=O(n) and bounded girth.