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

Every self-similar group acts on the space X_ω of infinite words over some alphabet X. We study the Schreier graphs Γ_w for w 2 X^ω of the action of self-similar groups generated by bounded automata on the space X^ω. Using sofic subshifts we determine the number of ends for every Schreier graph Γ_w. Almost all Schreier graphs Γ^w with respect to the uniform measure on X^w 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. [ABSTRACT FROM AUTHOR]