{
  "id": "1712.08210",
  "version": "v1",
  "published": "2017-12-21T20:56:17.000Z",
  "updated": "2017-12-21T20:56:17.000Z",
  "title": "A nonamenable \"factor\" of a euclidean space",
  "authors": [
    "Adam Timar"
  ],
  "comment": "22 pages, 4 figures",
  "categories": [
    "math.PR",
    "math.DS",
    "math.GR"
  ],
  "abstract": "Answering a question of Benjamini, we present an isometry-invariant random partition of the euclidean space R^3 into infinite connected indinstinguishable pieces, such that the adjacency graph defined on the pieces is the 3-regular infinite tree. Along the way, it is proved that any finitely generated amenable Cayley graph (or more generally, amenable unimodular random graph) can be represented in R^3 as an isometry-invariant random collection of polyhedral domains (tiles). A new technique is developed to prove indistinguishability for certain constructions, connecting this notion to factor of iid's.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-12-21T20:56:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "euclidean space",
      "isometry-invariant random partition",
      "isometry-invariant random collection",
      "amenable unimodular random graph",
      "finitely generated amenable cayley graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}