{
  "id": "math/0701225",
  "version": "v2",
  "published": "2007-01-08T13:46:32.000Z",
  "updated": "2007-12-26T14:34:48.000Z",
  "title": "Generation Gaps and Abelianised Defects of Free Products",
  "authors": [
    "Karl W. Gruenberg",
    "Peter A. Linnell"
  ],
  "comment": "18 pages, minor changes. To appear in J. Group Theory",
  "categories": [
    "math.GR"
  ],
  "abstract": "Let G be a group of the form G_1* ... *G_n, the free product of n subgroups, and let M be a ZG-module of the form $\\bigoplus_{i=1}^n M_i \\otimes_{\\mathbb{Z}G_i} \\mathbb{Z}G$. We shall give formulae in various situations for $d_{ZG}(M)$, the minimum number of elements required to generate M. In particular if C_1,C_2 are non-trivial finite cyclic groups of coprime orders, $G = (C_1 \\times Z) * (C_2 \\times Z)$ and $F/R \\cong G$ is the free presentation obtained from the natural free presentations of the two factors, then the number of generators of the relation module, $d_{\\mathbb{Z}G}(R/R')$ is three. It seems plausible that the minimum number of relators of G should be 4, and this would give a finitely presented group with positive relation gap. However we cannot prove this last statement.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2007-12-26T14:34:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F05",
      "20E06"
    ],
    "keywords": [
      "free product",
      "generation gaps",
      "abelianised defects",
      "minimum number",
      "non-trivial finite cyclic groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007math......1225G"
    }
  }
}