{
  "id": "1307.2593",
  "version": "v2",
  "published": "2013-07-09T20:44:32.000Z",
  "updated": "2015-04-09T12:16:11.000Z",
  "title": "Arithmetic quotients of the mapping class group",
  "authors": [
    "Fritz Grunewald",
    "Michael Larsen",
    "Alexander Lubotzky",
    "Justin Malestein"
  ],
  "comment": "46 pages, 1 figure, minor edits, added some references and changed the discussion of some other references",
  "categories": [
    "math.GT",
    "math.GR",
    "math.RT"
  ],
  "abstract": "To every $Q$-irreducible representation $r$ of a finite group $H$, there corresponds a simple factor $A$ of $Q[H]$ with an involution $\\tau$. To this pair $(A,\\tau)$, we associate an arithmetic group $\\Omega$ consisting of all $(2g-2)\\times (2g-2)$ matrices over a natural order of $A^{op}$ which preserve a natural skew-Hermitian sesquilinear form on $A^{2g-2}$. We show that if $H$ is generated by less than $g$ elements, then $\\Omega$ is a virtual quotient of the mapping class group $Mod(\\Sigma_g)$, i.e. a finite index subgroup of $\\Omega$ is a quotient of a finite index subgroup of $\\Mod(\\Sigma_g)$. This shows that the mapping class group has a rich family of arithmetic quotients (and \"Torelli subgroups\") for which the classical quotient $Sp(2g, Z)$ is just a first case in a list, the case corresponding to the trivial group $H$ and the trivial representation. Other pairs of $H$ and $r$ give rise to many new arithmetic quotients of $Mod(\\Sigma_g)$ which are defined over various (subfields of) cyclotomic fields and are of type $Sp(2m), SO(2m,2m),$ and $SU(m,m)$ for arbitrarily large $m$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-07-09T20:44:32.000Z",
      "comment": "44 pages, 1 figure",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-04-09T12:16:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M10",
      "57N05",
      "20G05"
    ],
    "keywords": [
      "mapping class group",
      "arithmetic quotients",
      "finite index subgroup",
      "natural skew-hermitian sesquilinear form",
      "cyclotomic fields"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 46,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1307.2593G"
    }
  }
}