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

2016/10/30 by Abdelrhman Elkasapy, Elkasapy, Abdelrhman · 1 citation
Mathematics · Physics and Astronomy · #Electromagnetic Scattering and Analysis #FOS: Mathematics #Group Theory (math.GR) #math.GR

paper · pdf · doi:10.48550/arxiv.1610.09725

7 pages

arxiv created 2016/10/30 · openalex publication_date 2016/10/30 · arxiv updated 2016/11/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We provide a new upper bound for the length for the shortest non-trivial element in the lower central series γ_n(\mathbbF₂) 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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