×

Ends of Schreier graphs and cut-points of limit spaces of self-similar groups. (English) Zbl 1423.20041

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.

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
PDFBibTeX XMLCite
Full Text: DOI arXiv