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

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.