{
  "id": "1807.11449",
  "version": "v1",
  "published": "2018-07-30T17:11:05.000Z",
  "updated": "2018-07-30T17:11:05.000Z",
  "title": "Non-residually finite extensions of arithmetic groups",
  "authors": [
    "Richard Hill"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "The aim of the article is to show that there are many finite extensions of arithmetic groups which are not residually finite. Suppose $G$ is a simple algebraic group over the rational numbers satisfying both strong approximation, and the congruence subgroup problem. We show that every arithmetic subgroup of $G$ has finite extensions which are not residually finite. More precisely, we investigate the group \\[ \\bar H^2(\\mathbb{Z}/n) = direct limit ( H^2(\\Gamma,\\mathbb{Z}/n) ), \\] where $\\Gamma$ runs through the arithmetic subgroups of $G$. Elements of $\\bar H^2(\\mathbb{Z}/n)$ correspond to (equivalence classes of) central extensions of arithmetic groups by $\\mathbb{Z}/n$; non-zero elements correspond to extensions which are not residually finite. We prove that $\\bar H^2(\\mathbb{Z}/n)$ contains infinitely many elements of order $n$, some of which are invariant for the action of the arithmetic completion $\\widehat{G(\\mathbb{Q})}$ of $G(\\mathbb{Q})$. We also investigate which of these (equivalence classes of) extensions lift to characteristic zero, by determining the invariant elements in the group \\[ \\bar H^2(\\mathbb{Z}_l) = projective limit \\bar H^2(\\mathbb{Z}/l^t). \\] We show that $\\bar H^2(\\mathbb{Z}_l)^{\\widehat{G(\\mathbb{Q})}}$ is isomorphic to $\\mathbb{Z}_l^c$ for some positive integer $c$. When $G(\\mathbb{R})$ has no simple components of complex type, we prove that $c=b+m$, where $b$ is the number of simple components of $G(\\mathbb{R})$ and $m$ is the dimension of the centre of a maximal compact subgroup of $G(\\mathbb{R})$. In all other cases, we prove upper and lower bounds on $c$; our lower bound (which we believe is the correct number) is $b+m$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-07-30T17:11:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F75"
    ],
    "keywords": [
      "arithmetic groups",
      "non-residually finite extensions",
      "simple components",
      "arithmetic subgroup",
      "equivalence classes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}