{
  "id": "1511.07418",
  "version": "v1",
  "published": "2015-11-23T20:56:02.000Z",
  "updated": "2015-11-23T20:56:02.000Z",
  "title": "A family of class-2 nilpotent groups, their automorphisms and pro-isomorphic zeta functions",
  "authors": [
    "Mark N. Berman",
    "Benjamin Klopsch",
    "Uri Onn"
  ],
  "comment": "21 pages",
  "categories": [
    "math.GR"
  ],
  "abstract": "The pro-isomorphic zeta function of a finitely generated nilpotent group $\\Gamma$ is a Dirichlet generating function that enumerates finite-index subgroups whose profinite completion is isomorphic to that of $\\Gamma$. Such zeta functions can be expressed as Euler products of $p$-adic integrals over the $p$-adic points of an algebraic automorphism group associated to $\\Gamma$. In this way they are closely related to classical zeta functions of algebraic groups over local fields. We describe the algebraic automorphism groups for a natural family of class-$2$ nilpotent groups; these groups can be viewed as generalizations of $D^*$-groups of odd Hirsch length. General $D^*$-groups, that is `indecomposable' finitely generated, torsion-free class-$2$ nilpotent groups with central Hirsch length $2$, were classified up to commensurability by Grunewald and Segal. We calculate the local pro-isomorphic zeta functions for our groups and obtain, in particular, explicit formulae for the local pro-isomorphic zeta functions associated to $D^*$-groups of odd Hirsch length. The global abscissae of convergence of the pro-isomorphic zeta functions of $D^*$-groups of odd Hirsch length are determined.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-11-23T20:56:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M41",
      "20E07",
      "20F18",
      "20F69",
      "17B40",
      "17B45"
    ],
    "keywords": [
      "nilpotent group",
      "odd hirsch length",
      "local pro-isomorphic zeta functions",
      "algebraic automorphism group",
      "central hirsch length"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151107418B"
    }
  }
}