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

Abdelrhman Elkasapy

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

Group Theory 2016-11-01 v1

Authors: Abdelrhman Elkasapy

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.

Keywords

lower central series free group shortest non-trivial element upper bound asymptotic behavior group laws compact groups

Cite

@article{arxiv.1610.09725,
  title  = {A new construction for the shortest non-trivial element in the lower central series},
  author = {Abdelrhman Elkasapy},
  journal= {arXiv preprint arXiv:1610.09725},
  year   = {2016}
}

Comments

7 pages

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