On a family of Schreier graphs of intermediate growth associated with a self-similar group

TL;DR

This paper constructs a family of infinite 4-regular Schreier graphs from self-similar group actions, characterizes their automorphisms, and proves they all exhibit intermediate growth, enriching understanding of graph growth behaviors.

Contribution

It introduces a new family of Schreier graphs associated with self-similar groups, solves their isomorphism problems, and establishes their intermediate growth property.

Findings

Graphs are 4-regular Schreier graphs from self-similar groups

Automorphism groups of these graphs are determined

All graphs in the family have intermediate growth

Abstract

For every infinite sequence $\omega=x_1,x_2,...$, with $x_i\in\{0,1\}$, we construct an infinite 4-regular graph $X_{\omega}$. These graphs are precisely the Schreier graphs of the action of a certain self-similar group on the space $\{0,1\}^{\infty}$. We solve the isomorphism and local isomorphism problems for these graphs, and determine their automorphism groups. Finally, we prove that all graphs $X_\omega$ have intermediate growth.