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

TL;DR

This paper investigates the structure of Schreier graphs generated by self-similar groups acting on infinite words, determining the number of ends and their relation to cut-points in the associated limit spaces.

Contribution

It introduces a method using sofic subshifts to classify the ends of Schreier graphs and characterizes automata whose graphs have two ends almost surely.

Findings

Most Schreier graphs have one or two ends.

A characterization of automata with almost surely two-ended Schreier graphs.

Established a connection between graph ends and cut-points in limit spaces.

Abstract

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.