Ends of Schreier graphs and cut-points of limit spaces of self-similar groups

Abstract

Every self-similar group acts on the space X^\omega of infinite words over some alphabet X . We study the Schreier graphs \Gamma_w for w\in X^\omega of the action of self-similar groups generated by bounded automata on the space X^\omega . Using sofic subshifts we determine the number of ends for every Schreier graph \Gamma_w . Almost all Schreier graphs \Gamma_w with respect to the uniform measure on X^\omega have one or two ends, and we characterize bounded automata whose Schreier graphs have two ends almost surely. The connection with (local) cut-points of limit spaces of self-similar groups is established.