{
  "id": "1002.0320",
  "version": "v2",
  "published": "2010-02-01T20:01:54.000Z",
  "updated": "2010-07-01T11:43:15.000Z",
  "title": "Finite generation of iterated wreath products",
  "authors": [
    "Ievgen Bondarenko"
  ],
  "comment": "7 pages. An example of a branch group with maximal subgroups of infinite index is added",
  "categories": [
    "math.GR"
  ],
  "abstract": "Let $(G_n,X_n)$ be a sequence of finite transitive permutation groups with uniformly bounded number of generators. We prove that the infinitely iterated permutational wreath product $...\\wr G_2\\wr G_1$ is topologically finitely generated if and only if the profinite abelian group $\\prod_{n\\geq 1} G_n/G'_n$ is topologically finitely generated. As a corollary, for a finite transitive group $G$ the minimal number of generators of the wreath power $G\\wr...\\wr G\\wr G$ ($n$ times) is bounded if $G$ is perfect, and grows linearly if $G$ is non-perfect. As a by-product we construct a finitely generated branch group, which has maximal subgroups of infinite index, answering [2,Question 14].",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-07-01T11:43:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F05",
      "20E22",
      "20E18",
      "20E08"
    ],
    "keywords": [
      "iterated wreath products",
      "finite generation",
      "profinite abelian group",
      "finite transitive permutation groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1002.0320B"
    }
  }
}