{
  "id": "2406.12506",
  "version": "v1",
  "published": "2024-06-18T11:15:08.000Z",
  "updated": "2024-06-18T11:15:08.000Z",
  "title": "Expanders and growth of normal subsets in finite simple groups of Lie type",
  "authors": [
    "Saveliy V. Skresanov"
  ],
  "comment": "13 pages",
  "categories": [
    "math.GR",
    "math.CO"
  ],
  "abstract": "We show that some classical results on expander graphs imply growth results on normal subsets in finite simple groups. As one application, it is shown that given a nontrivial normal subset $ A $ of a finite simple group $ G $ of Lie type of bounded rank, we either have $ G \\setminus \\{ 1 \\} \\subseteq A^2 $ or $ |A^2| \\geq |A|^{1+\\epsilon} $, for $ \\epsilon > 0 $. This improves a result of Gill, Pyber, Short and Szab\\'o, and partially resolves a question of Pyber from the Kourovka notebook. We also propose a variant of Gowers' trick for two subsets, and give applications to products of large subsets in groups of Lie type, improving some results of Larsen, Shalev and Tiep.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-18T11:15:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20D06",
      "05C48",
      "20P05"
    ],
    "keywords": [
      "finite simple group",
      "lie type",
      "expander graphs imply growth results",
      "nontrivial normal subset",
      "application"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}