{
  "id": "1403.0688",
  "version": "v2",
  "published": "2014-03-04T05:21:54.000Z",
  "updated": "2015-01-29T01:57:28.000Z",
  "title": "Splitting in orbit equivalence, treeable groups, and the Haagerup property",
  "authors": [
    "Yoshikata Kida"
  ],
  "comment": "31 pages, Theorem 1.4 improved",
  "categories": [
    "math.GR",
    "math.DS",
    "math.OA"
  ],
  "abstract": "Let $G$ be a discrete countable group and $C$ its central subgroup with $G/C$ treeable. We show that for any treeable action of $G/C$ on a standard probability space $X$, the groupoid $G\\ltimes X$ is isomorphic to the direct product of $C$ and $(G/C)\\ltimes X$, through cohomology of groupoids. We apply this to show that any group in the minimal class of groups containing treeable groups and closed under taking direct products, commensurable groups and central extensions has the Haagerup property.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-03-04T05:21:54.000Z",
      "abstract": "Let $G$ be a discrete countable group and $C$ its central subgroup with $G/C$ treeable. We show that for any treeable action of $G/C$ on a standard probability space $X$, the groupoid $G\\ltimes X$ is isomorphic to the direct product of $C$ and $(G/C)\\ltimes X$, through cohomology of groupoids. We apply this to show that any group in the minimal class of groups containing treeable groups and closed under taking direct products and central extensions has the Haagerup property.",
      "comment": "30 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-01-29T01:57:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A20"
    ],
    "keywords": [
      "haagerup property",
      "orbit equivalence",
      "direct product",
      "standard probability space",
      "central subgroup"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1403.0688K"
    }
  }
}