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

We provide a new upper bound for the length for the shortest non-trivial element in the lower central series $γ_n(\mathbb{F}_2)$ of the free group on two generators. We prove that it has an asymptotic behaviour of the form $O(n^{\log_φ(2)})$, where $φ=1.618...$ 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.

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