On the rigidity of rank gradient in a group of intermediate growth

TL;DR

This paper studies the rigidity of the rank gradient property in a specific intermediate growth group, showing it exhibits a particular form of rigidity characterized by logarithmic functions.

Contribution

It introduces the concept of normal (f,g)-RG rigidity and demonstrates this property for a specific intermediate growth group, expanding understanding of rank gradient behavior.

Findings

The group $ extit{ extbf{G}}$ is normally $(f,g)$-RG rigid with $f(n)= ext{log}(n)$ and $g(n)= ext{log}( ext{log}(n))$.

The paper establishes a new rigidity property for groups of intermediate growth.

It advances the theoretical understanding of rank gradient in complex group structures.

Abstract

We introduce and investigate the rigidity property of rank gradient in the case of the group $\mathcal G$ of intermediate growth constructed by the first author. We show that $\mathcal G$ is normally $(f,g)$-RG rigid where $f(n)=\log(n)$ and $g(n) =\log(\log(n)).$