{
  "id": "1804.05375",
  "version": "v1",
  "published": "2018-04-15T16:02:42.000Z",
  "updated": "2018-04-15T16:02:42.000Z",
  "title": "Commutator Subgroups of Twin Groups and Grothendieck's Cartographical groups",
  "authors": [
    "Soumya Dey",
    "Krishnendu Gongopadhyay"
  ],
  "comment": "15 pages, 1 figure, comments are most welcome",
  "categories": [
    "math.GT",
    "math.GR"
  ],
  "abstract": "Let $TW_n$ be the twin group on $n$ arcs. The group $TW_{n+2}$ is isomorphic to Grothendieck's $n$-dimensional cartographical group $\\mathcal C_n$, $n \\geq 1$. In this paper we give a finite presentation of the commutator subgroup $TW_{n+2}'$. We further prove that the commutator subgroup $TW_{n+2}'$ has rank $2n-1$, $n \\geq 1$. As corollaries, we derive that $TW_{n+2}'$ is free if and only if $n \\leq 3$. From this it follows that the automorphism group of $TW_{n+2}$ is finitely presented for $n \\leq 3$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-04-15T16:02:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F36",
      "20F12",
      "20F05",
      "11G32",
      "05E15"
    ],
    "keywords": [
      "commutator subgroup",
      "grothendiecks cartographical groups",
      "twin group",
      "automorphism group",
      "dimensional cartographical group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}