Uniform Diameter Bounds in Branch Groups

TL;DR

This paper establishes new polylogarithmic upper bounds on the diameters of quotients of specific branch groups, independent of generating sets, using profinite techniques and the groups' branch structures.

Contribution

It provides explicit, generating-set-independent diameter bounds for quotients of Grigorchuk and Gupta-Sidki groups, advancing understanding of their finite quotients.

Findings

Diameter bounds are polylogarithmic in group order.

Bounds are independent of generating sets.

Utilizes profinite Solovay-Kitaev procedure and branch structures.

Abstract

Let $G$ be either the Grigorchuk $2$-group or one of the Gupta-Sidki $p$-groups. We give new upper bounds for the diameters of the quotients of $G$ by its level stabilisers, as well as other natural sequences of finite-index normal subgroups. Our bounds are independent of the generating set, and are polylogarithmic functions of the group order, with explicit degree. Our proofs utilize a version of the profinite Solovay-Kitaev procedure, the branch structure of $G$, and in certain cases, results on the lower central series of $G$.