{
  "id": "1403.3277",
  "version": "v2",
  "published": "2014-03-13T14:28:01.000Z",
  "updated": "2014-10-06T14:09:25.000Z",
  "title": "Non-abelian group structure on the Urysohn space",
  "authors": [
    "Michal Doucha"
  ],
  "comment": "Substantially different version based on the referee's comments",
  "categories": [
    "math.GN",
    "math.LO"
  ],
  "abstract": "Following the continuing interest in the Urysohn space and, more specifically, the recent problem area of finding and comparing group structures on the Urysohn space we prove that there exists a non-abelian group structure on the Urysohn universal metric space. More precisely, we introduce a variant of the Graev metric that enables us to construct a free group with countably many generators equipped with a two-sided invariant metric that is isometric to the rational Urysohn space. We provide several open questions and problems related to this subject.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-03-13T14:28:01.000Z",
      "abstract": "Following the continuing interest in the Urysohn space and, more specifically, the recent problem area of finding and comparing group structures on the Urysohn space we prove that there exists a non-abelian group structure on the Urysohn universal metric space. More precisely, using the Fra\\\" iss\\' e theory we construct a free group with countably many generators equipped with a two-sided invariant metric that is isometric to the rational Urysohn space. We provide several open questions and problems related to this subject.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-10-06T14:09:25.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "non-abelian group structure",
      "urysohn universal metric space",
      "rational urysohn space",
      "comparing group structures",
      "open questions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1403.3277D"
    }
  }
}