On profinite groups with positive rank gradient

TL;DR

This paper demonstrates that profinite groups with positive rank gradient do not satisfy any group law, and in the pro-p case, contain a dense free subgroup, revealing new structural properties of such groups.

Contribution

It establishes that positive rank gradient implies the absence of group laws and the existence of dense free subgroups in pro-p groups, advancing understanding of their structure.

Findings

Profinite groups with positive rank gradient do not satisfy any group law.

Pro-p groups with positive rank gradient contain nonabelian dense free subgroups.

The results connect rank gradient with structural properties like free subgroups.

Abstract

We prove that a profinite group $G$ with positive rank gradient does not satisfy a group law. In the case when $G$ is a pro-$p$ group we show that $G$ contains a nonabelian dense free subgroup.