{
  "id": "2103.10991",
  "version": "v1",
  "published": "2021-03-19T19:18:26.000Z",
  "updated": "2021-03-19T19:18:26.000Z",
  "title": "Universal minimal flows of extensions of and by compact groups",
  "authors": [
    "Dana Bartošová"
  ],
  "comment": "11 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "Every topological group $G$ has up to isomorphism a unique minimal $G$-flow that maps onto every minimal $G$-flow, the universal minimal flow $M(G).$ We show that if $G$ has a compact normal subgroup $K$ that acts freely on $M(G)$ and there exists a uniformly continuous cross section $G/K\\to G,$ then the phase space of $M(G)$ is homeomorphic to the product of the phase space of $M(G/K)$ with $K$. Moreover, if either the left and right uniformities on $G$ coincide or $G\\cong G/K\\ltimes K$, we also recover the action, in the latter case extending a result of Kechris and Soki\\'c. As an application, we show that the phase space of $M(G)$ for any totally disconnected locally compact Polish group $G$ with a normal open compact subgroup is homeomorphic to a finite set, Cantor set $2^{\\mathbb{N}}$, $M(\\mathbb{Z})$, or $M(\\mathbb{Z})\\times 2^{\\mathbb{N}}.$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-03-19T19:18:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37B05",
      "22B99"
    ],
    "keywords": [
      "universal minimal flow",
      "compact groups",
      "phase space",
      "locally compact polish group",
      "disconnected locally compact polish"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}