{
  "id": "1112.6415",
  "version": "v2",
  "published": "2011-12-29T20:40:20.000Z",
  "updated": "2013-08-06T16:39:12.000Z",
  "title": "Quasi-actions and rough Cayley graphs for locally compact groups",
  "authors": [
    "Pekka Salmi"
  ],
  "comment": "15 pages; v2 is a major revision with simplifications and additions",
  "categories": [
    "math.GR"
  ],
  "abstract": "We define the notion of rough Cayley graph for compactly generated locally compact groups in terms of quasi-actions. We construct such a graph for any compactly generated locally compact group using quasi-lattices and show uniqueness up to quasi-isometry. A class of examples is given by the Cayley graphs of cocompact lattices in compactly generated groups. As an application, we show that a compactly generated group has polynomial growth if and only if its rough Cayley graph has polynomial growth (same for intermediate and exponential growth). Moreover, a unimodular compactly generated group is amenable if and only if its rough Cayley graph is amenable as a metric space.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-08-06T16:39:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C25",
      "20F65",
      "22D05",
      "43A07"
    ],
    "keywords": [
      "rough cayley graph",
      "compactly generated locally compact group",
      "compactly generated group",
      "quasi-actions",
      "polynomial growth"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1112.6415S"
    }
  }
}