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.