New Uniform Diameter Bounds in Pro-$p$ Groups

TL;DR

This paper establishes new, uniform upper bounds on the diameters of certain finite groups, independent of generating sets, using methods inspired by quantum computation and commutator structures.

Contribution

It introduces a novel approach to bounding group diameters that applies to various pro-$p$ groups and their quotients, extending previous results with polylogarithmic bounds.

Findings

Polylogarithmic diameter bounds for finite quotients of pro-$p$ groups.

Bounds applicable to Chevalley groups over pro-$p$ domains.

Implications for random walks on these groups.

Abstract

We give new upper bounds for the diameters of finite groups which do not depend on a choice of generating set. Our method exploits the commutator structure of certain profinite groups, in a fashion analogous to the Solovay-Kitaev procedure from quantum computation. We obtain polylogarithmic upper bounds for the diameters of finite quotients of: groups with an analytic structure over a pro-$p$ domain (with exponent depending on the dimension); Chevalley groups over a pro-$p$ domain (with exponent independent of the dimension) and the Nottingham group of a finite field. We also discuss some consequences of our results for random walks on groups.