{
  "id": "1304.2839",
  "version": "v2",
  "published": "2013-04-10T03:54:24.000Z",
  "updated": "2014-02-27T00:34:22.000Z",
  "title": "Amenability and Unique Ergodicity of Automorphism Groups of Fraïssé Structures",
  "authors": [
    "Andy Zucker"
  ],
  "categories": [
    "math.LO",
    "math.CO",
    "math.DS",
    "math.PR"
  ],
  "abstract": "In this paper we provide a necessary and sufficient condition for the amenability of the automorphism group of Fra\\\"iss\\'e structures and apply it to prove the non-amenability of the automorphism groups of the directed graph $\\mathbf{S}(3)$ and the Boron tree structure $\\mathbf{T}$. Also, we provide a negative answer to the Unique Ergodicity-Generic Point problem of Angel-Kechris-Lyons [AKL]. By considering $\\mathrm{GL}(\\mathbf{V}_\\infty)$, where $\\mathbf{V}_\\infty$ is the countably infinite dimensional vector space over a finite field $F_q$, we show that the unique invariant measure on the universal minimal flow of $\\mathrm{GL}(\\mathbf{V}_\\infty)$ is not supported on the generic orbit.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-02-27T00:34:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37B05"
    ],
    "keywords": [
      "automorphism group",
      "amenability",
      "countably infinite dimensional vector space",
      "unique ergodicity-generic point problem",
      "universal minimal flow"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1304.2839Z"
    }
  }
}