{
  "id": "1711.08295",
  "version": "v1",
  "published": "2017-11-22T14:34:58.000Z",
  "updated": "2017-11-22T14:34:58.000Z",
  "title": "Scaling limits of Cayley graphs with polynomially growing balls",
  "authors": [
    "Romain Tessera",
    "Matthew Tointon"
  ],
  "comment": "37 pages",
  "categories": [
    "math.GR",
    "math.CO",
    "math.GN",
    "math.MG"
  ],
  "abstract": "Benjamini, Finucane and the first author have shown that if (G_n,S_n) is a sequence of Cayley graphs such that |S_n^n|=O(n^D|S_n|), then the sequence (G_n,d_{S_n}/n) is relatively compact for the Gromov-Hausdorff topology and every cluster point is a connected nilpotent Lie group equipped with a left-invariant sub-Finsler metric. In this paper we show that the dimension of such a cluster point is bounded by D, and that, under the stronger bound |S_n^n|=O(n^D), the homogeneous dimension of a cluster point is bounded by D. Our approach is roughly to use a well-known structure theorem for approximate groups due to Breuillard, Green and Tao to replace S_n^n with a coset nilprogression of bounded rank, and then to use results about nilprogressions from a previous paper of ours to study the ultralimits of such coset nilprogressions. As an application we bound the dimension of the scaling limit of a sequence of vertex-transitive graphs of large diameter. We also recover and effectivise parts of an argument of Tao concerning the further growth of single set S satisfying the bound |S^n| < Mn^D|S|.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-11-22T14:34:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "polynomially growing balls",
      "cayley graphs",
      "scaling limit",
      "cluster point",
      "coset nilprogression"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}