{
  "id": "2207.10509",
  "version": "v1",
  "published": "2022-07-21T14:43:20.000Z",
  "updated": "2022-07-21T14:43:20.000Z",
  "title": "Model geometries of finitely generated groups",
  "authors": [
    "Alex Margolis"
  ],
  "comment": "39 pages. Comments welcome",
  "categories": [
    "math.GR"
  ],
  "abstract": "We study model geometries of finitely generated groups. If a finitely generated group does not contain a non-trivial finite rank free abelian commensurated subgroup, we show any model geometry is dominated by either a symmetric space of non-compact type, an infinite locally finite vertex-transitive graph, or a product of such spaces. We also prove that a finitely generated group possesses a model geometry not dominated by a locally finite graph if and only if it contains either a commensurated finite rank free abelian subgroup, or a uniformly commensurated subgroup that is a uniform lattice in a semisimple Lie group. This characterises finitely generated groups that embed as uniform lattices in locally compact groups that are not compact-by-(totally disconnected). We show the only such groups of cohomological two are surface groups and generalised Baumslag-Solitar groups, and we obtain an analogous characterisation for groups of cohomological dimension three.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-21T14:43:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F65",
      "20F67",
      "57M07"
    ],
    "keywords": [
      "finitely generated group",
      "model geometry",
      "free abelian commensurated subgroup",
      "locally finite vertex-transitive graph",
      "non-trivial finite rank free abelian"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 39,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}