{
  "id": "1408.4107",
  "version": "v1",
  "published": "2014-08-18T19:50:36.000Z",
  "updated": "2014-08-18T19:50:36.000Z",
  "title": "Automorphism groups of countable algebraically closed graphs and endomorphisms of the random graph",
  "authors": [
    "Igor Dolinka",
    "Robert D. Gray",
    "Jillian D. McPhee",
    "James D. Mitchell",
    "Martyn Quick"
  ],
  "comment": "26 pages, 3 figures",
  "categories": [
    "math.CO",
    "math.GR",
    "math.LO"
  ],
  "abstract": "We establish links between countable algebraically closed graphs and the endomorphisms of the countable universal graph $R$. As a consequence we show that, for any countable graph $\\Gamma$, there are uncountably many maximal subgroups of the endomorphism monoid of $R$ isomorphic to the automorphism group of $\\Gamma$. Further structural information about End $R$ is established including that Aut $\\Gamma$ arises in uncountably many ways as a Sch\\\"{u}tzenberger group. Similar results are proved for the countable universal directed graph and the countable universal bipartite graph.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-08-18T19:50:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C25",
      "03C50",
      "20M20",
      "20B27"
    ],
    "keywords": [
      "countable algebraically closed graphs",
      "automorphism group",
      "random graph",
      "endomorphism",
      "countable universal bipartite graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1408.4107D"
    }
  }
}