{
  "id": "1712.06727",
  "version": "v1",
  "published": "2017-12-19T00:38:18.000Z",
  "updated": "2017-12-19T00:38:18.000Z",
  "title": "On parabolic subgroups of Artin-Tits groups of spherical type",
  "authors": [
    "María Cumplido",
    "Volker Gebhardt",
    "Juan González-Meneses",
    "Bert Wiest"
  ],
  "comment": "35 pages, 8 figures",
  "categories": [
    "math.GR",
    "math.GT"
  ],
  "abstract": "We show that, in an Artin-Tits group of spherical type, the intersection of two parabolic subgroups is a parabolic subgroup. Moreover, we show that the set of parabolic subgroups forms a lattice with respect to inclusion. This extends to all Artin-Tits groups of spherical type a result that was previously known for braid groups. To obtain the above results, we show that every element in an Artin-Tits group of spherical type admits a unique minimal parabolic subgroup containing it. Also, the subgroup associated to an element coincides with the subgroup associated to any of its powers or roots. As a consequence, if an element belongs to a parabolic subgroup, all its roots belong to the same parabolic subgroup.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-12-19T00:38:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F36",
      "20F65"
    ],
    "keywords": [
      "artin-tits group",
      "spherical type",
      "parabolic subgroups forms",
      "unique minimal parabolic subgroup containing",
      "braid groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}