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

摘要

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(k)(n) (Z) (k >= 1). We also construct self-similar groups, whose graphs Gamma(n) can be represented as an iterated replacement product and graph powering: Gamma(n+1) = Gamma(k)(n) (r) (k >= 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 diam(Gamma(n)) = O(n) and bounded girth. (C) 2015 Elsevier Inc. All rights reserved.