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

2016/01/27 by Ievgen Bondarenko, Daniele D'Angeli, Tatiana Nagnibeda · 1 citation
Mathematics · #math.GR #math.CO #msc:20F65 #msc:05C63 #msc:05C25

paper · pdf · doi:10.4171/jfg/55

published as Journal of Fractal Geometry, Volume 4, Issue 4, 2017, 369-424

arxiv created 2016/01/27 · arxiv updated 2018/11/02

Abstract

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∈ 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^ω 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.

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