Journal of Fractal Geometry


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Volume 4, Issue 4, 2017, pp. 369–424
DOI: 10.4171/JFG/55

Published online: 2017-12-05

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

Ievgen Bondarenko[1], Daniele D'Angeli[2] and Tatiana Nagnibeda[3]

(1) National Taras Shevchenko University of Kyiv, Ukraine
(2) Technische Universität Graz, Austria
(3) Université de Genève, Switzerland

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.

Keywords: Self-similar group, Schreier graph, end of graph, bounded automaton, limit space, tile, cut-point

Bondarenko Ievgen, D'Angeli Daniele, Nagnibeda Tatiana: Ends of Schreier graphs and cut-points of limit spaces of self-similar groups. J. Fractal Geom. 4 (2017), 369-424. doi: 10.4171/JFG/55