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Ends of Schreier graphs and cut-points of limit spaces of self-similar groups (1601.07587v1)
Published 27 Jan 2016 in math.GR and math.CO
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.