{
  "id": "1310.7214",
  "version": "v1",
  "published": "2013-10-27T17:03:09.000Z",
  "updated": "2013-10-27T17:03:09.000Z",
  "title": "From the Poincaré Theorem to generators of the unit group of integral group rings of finite groups",
  "authors": [
    "Eric Jespers",
    "Stanley Orlando Juriaans",
    "Ann Kiefer",
    "Antonio de Andrade e Silva",
    "Anotnio Calixto Souza Filho"
  ],
  "comment": "32 pages, accepted in Mathematics of Computations. arXiv admin note: substantial text overlap with arXiv:1205.1127",
  "categories": [
    "math.GR",
    "math.RA"
  ],
  "abstract": "We give an algorithm to determine finitely many generators for a subgroup of finite index in the unit group of an integral group ring $\\mathbb{Z} G$ of a finite nilpotent group $G$, this provided the rational group algebra $\\mathbb{Q} G$ does not have simple components that are division classical quaternion algebras or two-by-two matrices over a classical quaternion algebra with centre $\\mathbb{Q}$. The main difficulty is to deal with orders in quaternion algebras over the rationals or a quadratic imaginary extension of the rationals. In order to deal with these we give a finite and easy implementable algorithm to compute a fundamental domain in the hyperbolic three space $\\mathbb{H}^3$ (respectively hyperbolic two space $\\mathbb{H}^2$) for a discrete subgroup of ${\\rm PSL}_2(\\mathbb{C})$ (respectively ${\\rm PSL}_2(\\mathbb{R})$) of finite covolume. Our results on group rings are a continuation of earlier work of Ritter and Sehgal, Jespers and Leal.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-10-27T17:03:09.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "integral group ring",
      "unit group",
      "finite groups",
      "generators",
      "division classical quaternion algebras"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1310.7214J"
    }
  }
}