The Limit Space of Self-similar Groups and Schreier graphs

TL;DR

This paper explores the limit space of self-similar groups generated by automata acting on rooted trees, approximating it via Schreier graphs and providing a computational tool for their analysis.

Contribution

It introduces a method to compute Schreier graphs for automatic groups and analyzes their Gromov-Hausdorff limit space.

Findings

Schreier graphs can be effectively computed for various automata groups.

The limit space is characterized as the Gromov-Hausdorff limit of Schreier graphs.

A Wolfram language program facilitates the analysis of these graphs.

Abstract

The present paper investigates the limit $G$-space $\mathcal{J}_{G}$ generated by the self-similar action of automatic groups on a regular rooted tree. The limit space $\mathcal{J}_{G}$ is the Gromov-Hausdorff limit of the family of Schreier graphs $\Gamma_{n}$; therefore, $\mathcal{J}_{G}$ can be approximated by Schreier graphs on level $n$-th when $n$ tends to infinity. We propose a computer program whose code is written in Wolfram language computes the adjacency matrix of Schreier graph $\Gamma_{n}$ at each specified level of the regular rooted tree. In this paper, the Schreier graphs corresponding to each automatic group is computed by applying the program to some collection of automata groups, including classic automatic groups.