{
  "id": "1901.05723",
  "version": "v1",
  "published": "2019-01-17T10:41:18.000Z",
  "updated": "2019-01-17T10:41:18.000Z",
  "title": "Ergodicity and type of nonsingular Bernoulli actions",
  "authors": [
    "Michael Björklund",
    "Zemer Kosloff",
    "Stefaan Vaes"
  ],
  "categories": [
    "math.DS",
    "math.GR",
    "math.OA",
    "math.PR"
  ],
  "abstract": "We determine the Krieger type of nonsingular Bernoulli actions $G \\curvearrowright \\prod_{g \\in G} (\\{0,1\\},\\mu_g)$. When $G$ is abelian, we do this for arbitrary marginal measures $\\mu_g$. We prove in particular that the action is never of type II$_\\infty$ if $G$ is abelian and not locally finite, answering Krengel's question for $G = \\mathbb{Z}$. When $G$ is locally finite, we prove that type II$_\\infty$ does arise. For arbitrary countable groups, we assume that the marginal measures stay away from $0$ and $1$. When $G$ has only one end, we prove that the Krieger type is always I, II$_1$ or III$_1$. When $G$ has more than one end, we show that other types always arise. Finally, we solve the conjecture of [VW17] by proving that a group $G$ admits a Bernoulli action of type III$_1$ if and only if $G$ has nontrivial first $L^2$-cohomology.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-01-17T10:41:18.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "nonsingular bernoulli actions",
      "ergodicity",
      "krieger type",
      "marginal measures stay away",
      "locally finite"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}