Growth of groups with linear Schreier graphs

TL;DR

This paper presents a novel method for estimating the growth of finitely generated groups using Schreier graphs, leading to new examples of groups with intermediate growth rates of the form exp(n^α).

Contribution

It introduces a new approach to bound group growth and constructs the first simple groups with intermediate growth rates.

Findings

Established upper bounds of the form exp(n^α) for group growth.

Constructed new examples of simple groups with intermediate growth.

Provided a novel method using graphs of group actions.

Abstract

We introduce a new method of proving upper estimates of growth of finitely generated groups and constructing groups of intermediate growth using graphs of their actions. These estimates are of the form $\exp(n^\alpha)$ for some $\alpha<1$, and provide the first examples of such bounds for simple groups of intermediate growth.