{
  "id": "math/0406013",
  "version": "v3",
  "published": "2004-06-01T12:19:43.000Z",
  "updated": "2004-06-15T15:02:44.000Z",
  "title": "Growth rates of amenable groups",
  "authors": [
    "Goulnara Arzhantseva",
    "Victor Guba",
    "Luc Guyot"
  ],
  "comment": "6 pages",
  "categories": [
    "math.GR"
  ],
  "abstract": "Let $F_m$ be a free group with $m$ generators and let $R$ be its normal subgroup such that $F_m/R$ projects onto $\\zz$. We give a lower bound for the growth rate of the group $F_m/R'$ (where $R'$ is the derived subgroup of $R$) in terms of the length $\\rho=\\rho(R)$ of the shortest nontrivial relation in $R$. It follows that the growth rate of $F_m/R'$ approaches $2m-1$ as $\\rho$ approaches infinity. This implies that the growth rate of an $m$-generated amenable group can be arbitrarily close to the maximum value $2m-1$. This answers an open question by P. de la Harpe. In fact we prove that such groups can be found already in the class of abelian-by-nilpotent groups as well as in the class of finite extensions of metabelian groups.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2004-06-15T15:02:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F65",
      "05C25"
    ],
    "keywords": [
      "growth rate",
      "amenable group",
      "shortest nontrivial relation",
      "normal subgroup",
      "open question"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......6013A"
    }
  }
}