{
  "id": "1503.06947",
  "version": "v1",
  "published": "2015-03-24T08:40:40.000Z",
  "updated": "2015-03-24T08:40:40.000Z",
  "title": "Uniform analytic properties of representation zeta functions of finitely generated nilpotent groups",
  "authors": [
    "Duong Hoang Dung",
    "Christopher Voll"
  ],
  "comment": "21 pages",
  "categories": [
    "math.GR",
    "math.RT"
  ],
  "abstract": "Let $G$ be a finitely generated torsion-free nilpotent group. The representation zeta function $\\zeta_G(s)$ of $G$ enumerates twist isoclasses of finite-dimensional irreducible complex representations of $G$. We prove that $\\zeta_G(s)$ has rational abscissa of convergence $a(G)$ and may be meromorphically continued to the left of $a(G)$ and that, on the line $\\{s\\in\\mathbb{C} \\mid \\textrm{Re}(s) = a(G)\\}$, the continued function is holomorphic except for a pole at $s=a(G)$. A Tauberian theorem yields a precise asymptotic result on the representation growth of $G$ in terms of the position and order of this pole. We obtain these results as a consequence of a more general result establishing uniform analytic properties of representation zeta functions of finitely generated nilpotent groups of the form $\\mathbf{G}(\\mathcal{O})$, where $\\mathbf{G}$ is a unipotent group scheme defined in terms of a nilpotent Lie lattice over the ring $\\mathcal{O}$ of integers of a number field. This allows us to show, in particular, that the abscissae of convergence of the representation zeta functions of such groups and their pole orders are invariants of $\\mathbf{G}$, independent of $\\mathcal{O}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-03-24T08:40:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F18",
      "20E18",
      "22E55",
      "20F69",
      "11M41"
    ],
    "keywords": [
      "representation zeta function",
      "finitely generated nilpotent groups",
      "establishing uniform analytic properties",
      "result establishing uniform analytic",
      "generated torsion-free nilpotent group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150306947H"
    }
  }
}