A new construction for the shortest non-trivial element in the lower central series

TL;DR

This paper introduces a novel upper bound for the length of the shortest non-trivial element in the lower central series of a free group, with implications for laws in finite and compact groups.

Contribution

It presents a new construction that improves bounds on the minimal element length in the lower central series and applies this to laws in finite and compact groups.

Findings

Shortest non-trivial element length grows as O(n^{log_phi(2)})

New estimates for laws in finite groups

Enhanced bounds for almost laws in compact groups

Abstract

We provide a new upper bound for the length for the shortest non-trivial element in the lower central series $\gamma_n(\mathbb{F}_2)$ of the free group on two generators. We prove that it has an asymptotic behaviour of the form $O(n^{\log_{\varphi}(2)})$, where $\varphi=1.618...$ is the golden ratio. This new technique is used to provide new estimates on the length of laws for finite groups and on almost laws for compact groups.