{
  "id": "1109.2351",
  "version": "v3",
  "published": "2011-09-11T20:34:21.000Z",
  "updated": "2012-05-21T18:33:32.000Z",
  "title": "The Borel complexity of von Neumann equivalence",
  "authors": [
    "Inessa Epstein",
    "Asger Tornquist"
  ],
  "comment": "28 pages. Minor corrections throughout",
  "categories": [
    "math.DS",
    "math.GR",
    "math.LO",
    "math.OA"
  ],
  "abstract": "We prove that for a countable discrete group $\\Gamma$ containing a copy of the free group $\\F_n$, for some $2\\leq n\\leq\\infty$, as a normal subgroup, the equivalence relations of conjugacy, orbit equivalence and von Neumann equivalence of the ergodic a.e. free actions of $\\Gamma$ are analytic non-Borel equivalence relations in the Polish space of probability measure preserving $\\Gamma$ actions. As a consequence we obtain that the isomorphism relation in the spaces of separably acting factors of type $\\II_1$, $\\II_\\infty$ and $\\III_\\lambda$, $0\\leq\\lambda\\leq 1$, are analytic and not Borel when these spaces are given the Effros Borel structure.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2012-05-21T18:33:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E15",
      "28D15",
      "37A35",
      "37A20",
      "46L36"
    ],
    "keywords": [
      "von neumann equivalence",
      "borel complexity",
      "analytic non-borel equivalence relations",
      "effros borel structure",
      "free group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1109.2351E"
    }
  }
}