{
  "id": "0809.0131",
  "version": "v2",
  "published": "2008-08-31T15:20:53.000Z",
  "updated": "2010-02-22T20:59:09.000Z",
  "title": "Representation zeta functions of wreath products with finite groups",
  "authors": [
    "Laurent Bartholdi",
    "Pierre de la Harpe"
  ],
  "comment": "35 pages, amstex source",
  "journal": "Groups Geom. Dyn. 4 (2010), 209--249",
  "doi": "10.4171/GGD/81",
  "categories": [
    "math.GR"
  ],
  "abstract": "Let G be a group which has for all n a finite number r_n(G) of irreducible complex linear representations of dimension n. Let $\\zeta(G,s) = \\sum_{n=1}^{\\infty} r_n(G) n^{-s}$ be its representation zeta function. First, in case G is a permutational wreath product of H with a permutation group Q acting on a finite set X, we establish a formula for $\\zeta(G,s)$ in terms of the zeta functions of H and of subgroups of Q, and of the Moebius function associated with the lattice of partitions of X in orbits under subgroups of Q. Then, we consider groups W(Q,k) which are k-fold iterated wreath products of Q, and several related infinite groups W(Q), including the profinite group, a locally finite group, and several finitely generated groups, which are all isomorphic to a wreath product of themselves with Q. Under convenient hypotheses (in particular Q should be perfect), we show that r_n(W(Q)) is finite for all n, and we establish that the Dirichlet series $\\zeta(W(Q),s)$ has a finite and positive abscissa of convergence s_0. Moreover, the function $\\zeta(W(Q),s)$ satisfies a remarkable functional equation involving $\\zeta(W(Q),es)$ for e=1,...,|X|. As a consequence of this, we exhibit some properties of the function, in particular that $\\zeta(W(Q),s)$ has a singularity at s_0, a finite value at s_0, and a Puiseux expansion around s_0. We finally report some numerical computations for Q the simple groups of order 60 and 168.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-02-22T20:59:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M41",
      "20C15",
      "20E22"
    ],
    "keywords": [
      "representation zeta function",
      "k-fold iterated wreath products",
      "irreducible complex linear representations",
      "permutational wreath product",
      "related infinite groups"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "AMS-TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0809.0131B"
    }
  }
}