Bondarenko, Ievgen; D’Angeli, Daniele; Nagnibeda, Tatiana Ends of Schreier graphs and cut-points of limit spaces of self-similar groups. (English) Zbl 1423.20041 J. Fractal Geom. 4, No. 4, 369-424 (2017). 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. Cited in 10 Documents MSC: 20F65 Geometric group theory 05C25 Graphs and abstract algebra (groups, rings, fields, etc.) 05C63 Infinite graphs 20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects) 28A80 Fractals 68Q70 Algebraic theory of languages and automata Keywords:self-similar group; Schreier graph; end of graph; bounded automaton; limit space; tile; cut-point PDFBibTeX XMLCite \textit{I. Bondarenko} et al., J. Fractal Geom. 4, No. 4, 369--424 (2017; Zbl 1423.20041) Full Text: DOI arXiv