{
  "id": "1803.06583",
  "version": "v1",
  "published": "2018-03-17T22:53:48.000Z",
  "updated": "2018-03-17T22:53:48.000Z",
  "title": "Circular orders, ultrahomogeneity and topological groups",
  "authors": [
    "Eli Glasner",
    "Michael Megrelishvili"
  ],
  "comment": "17 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "We study topological groups $G$ for which the universal minimal $G$-system $M(G)$, or the universal irreducible affine $G$-system $IA(G)$ are tame. We call such groups intrinsically tame and convexly intrinsically tame. These notions are generalized versions of extreme amenability and amenability, respectively. When $M(G)$, as a $G$-system, admits a circular order we say that $G$ is intrinsically circularly ordered. This implies that $G$ is intrinsically tame. We show that for every circularly ultrahomogeneous action $G \\curvearrowright X$ on a circularly ordered set $X$ the topological group $G$, in its pointwise convergence topology, is intrinsically circularly ordered. This result is a \"circular\" analog of Pestov's result about the extremal amenability of ultrahomogeneous actions on linearly ordered sets by linear order preserving transformations. In the case where $X$ is countable, the corresponding Polish group of circular automorphisms $G$ admits a concrete description. Using the Kechris-Pestov-Todorcevic construction we show that $M(G)$ is a circularly ordered compact space obtained by splitting the rational points on the circle. We show also that $G$ is Roelcke precompact, satisfies Kazhdan's property $T$ (using results of Evans-Tsankov) and has the automatic continuity property (using results of Rosendal-Solecki).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-03-17T22:53:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37Bxx",
      "54H20",
      "54H15",
      "22A25"
    ],
    "keywords": [
      "topological group",
      "circular order",
      "ultrahomogeneity",
      "ultrahomogeneous action",
      "automatic continuity property"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}