{
  "id": "0711.3514",
  "version": "v1",
  "published": "2007-11-22T07:58:23.000Z",
  "updated": "2007-11-22T07:58:23.000Z",
  "title": "The ratio and generating function of cogrowth coefficients of finitely generated groups",
  "authors": [
    "Ryszard Szwarc"
  ],
  "journal": "Studia Mathematica 131 (1998), 89-94",
  "categories": [
    "math.FA",
    "math.GR"
  ],
  "abstract": "Let G be a group generated by $r$ elements $g_1,g_2,..., g_r.$ Among the reduced words in $g_1,g_2,..., g_r$ of length $n$ some, say $\\gamma_n,$ represent the identity element of the group $G.$ It has been shown in a combinatorial way that the $2n$th root of $\\gamma_{2n}$ has a limit, called the cogrowth exponent with respect to generators $g_1,g_2,..., g_r.$ We show by analytic methods that the numbers $\\gamma_n$ vary regularly; i.e. the ratio $\\gamma_{2n+2}/\\gamma_{2n}$ is also convergent. Moreover we derive new precise information on the domain of holomorphy of $\\gamma(z),$ the generating function associated with the coefficients $\\gamma_n.$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-11-22T07:58:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F05",
      "20E05"
    ],
    "keywords": [
      "finitely generated groups",
      "generating function",
      "cogrowth coefficients",
      "combinatorial way",
      "precise information"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0711.3514S"
    }
  }
}