{
  "id": "1705.03528",
  "version": "v1",
  "published": "2017-05-09T20:28:04.000Z",
  "updated": "2017-05-09T20:28:04.000Z",
  "title": "The space of stable weak equivalence classes of measure-preserving actions",
  "authors": [
    "Lewis Bowen",
    "Robin Tucker-Drob"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "The concept of (stable) weak containment for measure-preserving actions of a countable group $\\Gamma$ is analogous to the classical notion of (stable) weak containment of unitary representations. If $\\Gamma$ is amenable then the Rokhlin lemma shows that all essentially free actions are weakly equivalent. However if $\\Gamma$ is non-amenable then there can be many different weak and stable weak equivalence classes. Our main result is that the set of stable weak equivalence classes naturally admits the structure of a Choquet simplex. For example, when $\\Gamma=\\mathbb{Z}$ this simplex has only a countable set of extreme points but when $\\Gamma$ is a nonamenable free group, this simplex is the Poulsen simplex. We also show that when $\\Gamma$ contains a nonabelian free group, this simplex has uncountably many strongly ergodic essentially free extreme points.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-05-09T20:28:04.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "stable weak equivalence classes",
      "measure-preserving actions",
      "essentially free extreme points",
      "ergodic essentially free extreme",
      "weak containment"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}