Ends of Schreier graphs and cut-points of limit spaces of self-similar groups (Q1700654)

Summary: 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.