{
  "id": "1804.09207",
  "version": "v1",
  "published": "2018-04-24T18:39:45.000Z",
  "updated": "2018-04-24T18:39:45.000Z",
  "title": "Finitely $\\mathcal{F}$-amenable actions and Decomposition Complexity of Groups",
  "authors": [
    "Andrew Nicas",
    "David Rosenthal"
  ],
  "categories": [
    "math.GT",
    "math.GR"
  ],
  "abstract": "In his work on the Farrell-Jones Conjecture, Arthur Bartels introduced the concept of a \"finitely $\\mathcal{F}$-amenable\" group action, where $\\mathcal{F}$ is a family of subgroups. We show how a finitely $\\mathcal{F}$-amenable action of a finitely generated group $G$ on a compact metric space, where the asymptotic dimensions of the elements of $\\mathcal{F}$ are bounded from above, gives an upper bound for the asymptotic dimension of $G$. We generalize this to families $\\mathcal{F}$ whose elements are contained in a collection, $\\mathfrak{C}$, of metric families that satisfies some basic permanence properties: If $G$ is a finitely generated group and each element of $\\mathcal{F}$ belongs to $\\mathfrak{C}$ and there exists a finitely $\\mathcal{F}$-amenable action of $G$ on a compact metrizable space, then $G$ is in $\\mathfrak{C}$. Examples of such collections of metric families include: metric families with weak finite decomposition complexity, exact metric families, and metric families that coarsely embed into Hilbert space.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-04-24T18:39:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F69",
      "20F65"
    ],
    "keywords": [
      "amenable action",
      "weak finite decomposition complexity",
      "finitely generated group",
      "asymptotic dimension",
      "compact metric space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}