{
  "id": "1610.09725",
  "version": "v1",
  "published": "2016-10-30T22:59:35.000Z",
  "updated": "2016-10-30T22:59:35.000Z",
  "title": "A new construction for the shortest non-trivial element in the lower central series",
  "authors": [
    "Abdelrhman Elkasapy"
  ],
  "comment": "7 pages",
  "categories": [
    "math.GR"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-10-30T22:59:35.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "lower central series",
      "shortest non-trivial element",
      "construction",
      "upper bound",
      "compact groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}