{
  "id": "0704.0095",
  "version": "v2",
  "published": "2007-04-01T16:59:04.000Z",
  "updated": "2012-04-10T05:13:40.000Z",
  "title": "Geometry of Locally Compact Groups of Polynomial Growth and Shape of Large Balls",
  "authors": [
    "Emmanuel Breuillard"
  ],
  "comment": "slightly expanded and polished new version, 57 pages, 2 figures",
  "categories": [
    "math.GR",
    "math.DG"
  ],
  "abstract": "We get asymptotics for the volume of large balls in an arbitrary locally compact group G with polynomial growth. This is done via a study of the geometry of G and a generalization of P. Pansu's thesis. In particular, we show that any such G is weakly commensurable to some simply connected solvable Lie group S, the Lie shadow of G. We also show that large balls in G have an asymptotic shape, i.e. after a suitable renormalization, they converge to a limiting compact set which can be interpreted geometrically. We then discuss the speed of convergence, treat some examples and give an application to ergodic theory. We also answer a question of Burago about left invariant metrics and recover some results of Stoll on the irrationality of growth series of nilpotent groups.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-04-10T05:13:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E25",
      "20F18",
      "37A15",
      "53C17"
    ],
    "keywords": [
      "large balls",
      "polynomial growth",
      "left invariant metrics",
      "arbitrary locally compact group",
      "limiting compact set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 57,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0704.0095B"
    }
  }
}