# Journal of Fractal Geometry

Full-Text PDF (860 KB) | Metadata | Table of Contents | JFG summary

**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