{
  "id": "0810.2027",
  "version": "v2",
  "published": "2008-10-11T13:52:04.000Z",
  "updated": "2011-06-22T18:46:58.000Z",
  "title": "Normal Subgroups of Profinite Groups of Non-negative Deficiency",
  "authors": [
    "Fritz Grunewald",
    "Andrei Jaikin-Zapirain",
    "Aline G. S. Pinto",
    "Pavel A. Zalesski"
  ],
  "comment": "This is a revised version; we added a result stating that arithmetic Klenian groups are good",
  "categories": [
    "math.GR"
  ],
  "abstract": "We initiate the study of profinite groups of non-negative deficiency. The principal focus of the paper is to show that the existence of a finitely generated normal subgroup of infinite index in a profinite group $G$ of non-negative deficiency gives rather strong consequences for the structure of $G$. To make this precise we introduce the notion of $p$-deficiency ($p$ a prime) for a profinite group $G$. This concept is more useful in the study of profinite groups then the notion of deficiency. We prove that if the $p$-deficiency of $G$ is positive and $N$ is a finitely generated normal subgroup such that the $p$-Sylow subgroup of $G/N$ is infinite and $p$ divides the order of $N$ then we have $\\cd_p(G)=2$, $\\cd_p(N)=1$ and $\\vcd_p(G/N)=1$ for the cohomological $p$-dimensions; moreover either the $p$-Sylow subgroup of $G/N$ is virtually cyclic or the $p$-Sylow subgroup of $N$ is cyclic. A profinite Poincar\\'e duality group $G$ of dimension 3 at a prime $p$ ($PD^3$-group) has deficiency 0. In this case we show that for $N$ and $p$ as above either $N$ is $PD^1$ at $p$ and $G/N$ is virtually $PD^2$ at $p$ or $N$ is $PD^2$ at $p$ and $G/N$ is virtually $PD^1$ at $p$. In particular if $G$ is pro-$p$ then either $N$ is infinite cyclic and $G/N$ is virtually Demushkin or $N$ is Demushkin and $G/N$ is virtually infinite cyclic. We apply this results to deduce structural information on the profinite completions of ascending HNN-extensions of free groups. We also give some implications of our theory to the congruence kernels of certain arithmetic groups.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-06-22T18:46:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20E18"
    ],
    "keywords": [
      "profinite group",
      "non-negative deficiency",
      "sylow subgroup",
      "finitely generated normal subgroup",
      "infinite cyclic"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0810.2027G"
    }
  }
}