{
  "id": "2303.09804",
  "version": "v1",
  "published": "2023-03-17T07:19:15.000Z",
  "updated": "2023-03-17T07:19:15.000Z",
  "title": "Commutator subgroups and crystallographic quotients of virtual extensions of symmetric groups",
  "authors": [
    "Pravin Kumar",
    "Tushar Kanta Naik",
    "Neha Nanda",
    "Mahender Singh"
  ],
  "comment": "29 pages",
  "categories": [
    "math.GR",
    "math.GT"
  ],
  "abstract": "The virtual braid group $VB_n$, the virtual twin group $VT_n$ and the virtual triplet group $VL_n$ are extensions of the symmetric group $S_n$, which are motivated by the Alexander-Markov correspondence for virtual knot theories. The kernels of natural epimorphisms of these groups onto the symmetric group $S_n$ are the pure virtual braid group $VP_n$, the pure virtual twin group $PVT_n$ and the pure virtual triplet group $PVL_n$, respectively. In this paper, we investigate commutator subgroups, pure subgroups and crystallographic quotients of these groups. We derive explicit finite presentations of the pure virtual triplet group $PVL_n$, the commutator subgroup $VT_n^{'}$ of $VT_n$ and the commutator subgroup $VL_n^{'}$ of $VL_n$. Our results complete the understanding of these groups, except that of $VB_n^{'}$, for which the existence of a finite presentations is not known for $n \\ge 4$. We also prove that $VL_n/PVL_n^{'}$ is a crystallographic group and give an explicit construction of infinitely many torsion elements in it.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-03-17T07:19:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F55",
      "20H15",
      "20F36"
    ],
    "keywords": [
      "commutator subgroup",
      "symmetric group",
      "crystallographic quotients",
      "pure virtual triplet group",
      "virtual extensions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}