{
  "id": "0806.2788",
  "version": "v2",
  "published": "2008-06-17T13:52:03.000Z",
  "updated": "2009-02-18T07:42:41.000Z",
  "title": "Free products, Orbit Equivalence and Measure Equivalence Rigidity",
  "authors": [
    "Aurélien Alvarez",
    "Damien Gaboriau"
  ],
  "comment": "minor additions",
  "categories": [
    "math.GR",
    "math.OA"
  ],
  "abstract": "We study the analogue in orbit equivalence of free product decomposition and free indecomposability for countable groups. We introduce the (orbit equivalence invariant) notion of freely indecomposable ({\\FI}) standard probability measure preserving equivalence relations and establish a criterion to check it, namely non-hyperfiniteness and vanishing of the first $L^2$-Betti number. We obtain Bass-Serre rigidity results, \\textit{i.e.} forms of uniqueness in free product decompositions of equivalence relations with ({\\FI}) components. The main features of our work are weak algebraic assumptions and no ergodicity hypothesis for the components. We deduce, for instance, that a measure equivalence between two free products of non-amenable groups with vanishing first $\\ell^2$-Betti numbers is induced by measure equivalences of the components. We also deduce new classification results in Orbit Equivalence and II$_1$ factors.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2009-02-18T07:42:41.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "orbit equivalence",
      "measure equivalence rigidity",
      "free product decomposition",
      "probability measure preserving equivalence relations",
      "standard probability measure preserving equivalence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0806.2788A"
    }
  }
}