{
  "id": "1404.6091",
  "version": "v1",
  "published": "2014-04-24T11:23:50.000Z",
  "updated": "2014-04-24T11:23:50.000Z",
  "title": "Presentations for quaternionic $S$-unit groups",
  "authors": [
    "Ted Chinburg",
    "Holley Friedlander",
    "Sean Howe",
    "Michiel Kosters",
    "Bhairav Singh",
    "Matthew Stover",
    "Ying Zhang",
    "Paul Ziegler"
  ],
  "categories": [
    "math.NT",
    "math.GR",
    "math.GT",
    "math.RA"
  ],
  "abstract": "The purpose of this paper is to give presentations for projective $S$-unit groups of the Hurwitz order in Hamilton's quaternions over the rational field $\\mathbb{Q}$. To our knowledge, this provides the first explicit presentations of an $S$-arithmetic lattice in a semisimple Lie group with $S$ large. In particular, we give presentations for groups acting irreducibly and cocompactly on a product of Bruhat--Tits trees. We also include some discussion and experimentation related to the congruence subgroup problem, which is open when $S$ contains at least two odd primes. In the appendix, we provide code that allows the reader to compute presentations for an arbitrary finite set $S$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-04-24T11:23:50.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "unit groups",
      "quaternionic",
      "arbitrary finite set",
      "congruence subgroup problem",
      "first explicit presentations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1404.6091C"
    }
  }
}