{
  "id": "1804.03049",
  "version": "v2",
  "published": "2018-04-09T15:06:23.000Z",
  "updated": "2018-09-14T16:16:14.000Z",
  "title": "Growth in linear algebraic groups and permutation groups: towards a unified perspective",
  "authors": [
    "Harald A. Helfgott"
  ],
  "comment": "To appear in the proceedings of the St Andrews conference (2017). Minor corrections + grant information",
  "categories": [
    "math.GR"
  ],
  "abstract": "By now, we have a product theorem in every finite simple group $G$ of Lie type, with the strength of the bound depending only in the rank of $G$. Such theorems have numerous consequences: bounds on the diameters of Cayley graphs, spectral gaps, and so forth. For the alternating group Alt_n, we have a quasipolylogarithmic diameter bound (Helfgott-Seress 2014), but it does not rest on a product theorem. We shall revisit the proof of the bound for Alt_n, bringing it closer to the proof for linear algebraic groups, and making some common themes clearer. As a result, we will show how to prove a product theorem for Alt_n -- not of full strength, as that would be impossible, but strong enough to imply the diameter bound.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2018-09-14T16:16:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20B05",
      "20B30",
      "30B35"
    ],
    "keywords": [
      "linear algebraic groups",
      "permutation groups",
      "product theorem",
      "unified perspective",
      "quasipolylogarithmic diameter bound"
    ],
    "tags": [
      "conference paper"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}