An Example of an Automatic Graph of Intermediate Growth

TL;DR

This paper presents a specific example of an infinite automatic graph with intermediate growth, constructed as a Schreier graph of a group generated by a 3-state automaton, addressing an open problem in the field.

Contribution

It provides the first known example of a 4-regular infinite automatic graph of intermediate growth, constructed via a Schreier graph related to automaton-generated groups.

Findings

Constructed a 4-regular infinite automatic graph of intermediate growth

Used a Schreier graph of a 3-state automaton-generated group

Addresses an open problem on Cayley automatic groups of intermediate growth

Abstract

We give an example of a 4-regular infinite automatic graph of intermediate growth. It is constructed as a Schreier graph of a certain group generated by 3-state automaton. The question was motivated by an open problem on the existence of Cayley automatic groups of intermediate growth.