{
  "id": "2108.09009",
  "version": "v1",
  "published": "2021-08-20T05:44:28.000Z",
  "updated": "2021-08-20T05:44:28.000Z",
  "title": "$\\mathrm{L}^1$ full groups of flows",
  "authors": [
    "François Le Maître",
    "Konstantin Slutsky"
  ],
  "categories": [
    "math.DS",
    "math.FA",
    "math.GR"
  ],
  "abstract": "We introduce the concept of an $\\mathrm{L}^{1}$ full group associated with a measure-preserving action of a Polish normed group on a standard probability space. Such groups are shown to carry a natural separable complete metric, and are thus Polish. Our construction generalizes $\\mathrm{L}^{1}$ full groups of actions of discrete groups, which have been studied recently by the first author. We show that under minor assumptions on the actions, topological derived subgroups of $\\mathrm{L}^{1}$ full groups are topologically simple and - when the acting group is locally compact and amenable - are whirly amenable and generically two-generated. For measure-preserving actions of the real line (also known as measure-preserving flows), the topological derived subgroup of an $\\mathrm{L}^{1}$ full groups is shown to coincide with the kernel of the index map, which implies that $\\mathrm{L}^{1}$ full groups of free measure-preserving flows are topologically finitely generated if and only if the flow admits finitely many ergodic components. The latter is in a striking contrast to the case of $\\mathbb{Z}$-actions, where the number of topological generators is controlled by the entropy of the action.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-08-20T05:44:28.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "full group",
      "topological derived subgroup",
      "standard probability space",
      "measure-preserving action",
      "natural separable complete metric"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}