{
  "id": "1001.4556",
  "version": "v1",
  "published": "2010-01-25T22:17:12.000Z",
  "updated": "2010-01-25T22:17:12.000Z",
  "title": "Growth in finite simple groups of Lie type",
  "authors": [
    "László Pyber",
    "Endre Szabó"
  ],
  "comment": "Research announcement",
  "categories": [
    "math.GR",
    "math.CO"
  ],
  "abstract": "We prove that if L is a finite simple group of Lie type and A a symmetric set of generators of L, then A grows i.e |AAA| > |A|^(1+epsilon) where epsilon depends only on the Lie rank of L, or AAA=L. This implies that for a family of simple groups L of Lie type the diameter of any Cayley graph is polylogarithmic in |L|. Combining our result on growth with known results of Bourgain,Gamburd and Varj\\'u it follows that if LAMBDA is a Zariski-dense subgroup of SL(d,Z) generated by a finite symmetric set S, then for square-free moduli m which are relatively prime to some number m_0 the Cayley graphs Gamma(SL(d,m),pi_m(S)) form an expander family.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-01-25T22:17:12.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finite simple group",
      "lie type",
      "cayley graph",
      "finite symmetric set",
      "square-free moduli"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1001.4556P"
    }
  }
}