{
  "id": "1702.08358",
  "version": "v1",
  "published": "2017-02-27T16:29:23.000Z",
  "updated": "2017-02-27T16:29:23.000Z",
  "title": "The Markoff Group of Transformations in Prime and Composite Moduli",
  "authors": [
    "Chen Meiri",
    "Doron Puder",
    "Dan Carmon"
  ],
  "comment": "26 pages, by Chen Meiri and Doron Puder, with an appendix by Dan Carmon",
  "categories": [
    "math.NT",
    "math.GR"
  ],
  "abstract": "The Markoff group of transformations is a group $\\Gamma$ of affine integral morphisms, which is known to act transitively on the set of all positive integer solutions to the equation $x^{2}+y^{2}+z^{2}=xyz$. The fundamental strong approximation conjecture for the Markoff equation states that for every prime $p$, the group $\\Gamma$ acts transitively on the set $X^{*}\\left(p\\right)$ of non-zero solutions to the same equation over $\\mathbb{Z}/p\\mathbb{Z}$. Recently, Bourgain, Gamburd and Sarnak proved this conjecture for all primes outside a small exceptional set. In the current paper, we study a group of permutations obtained by the action of $\\Gamma$ on $X^{*}\\left(p\\right)$, and show that for most primes, it is the full symmetric or alternating group. We use this result to deduce that $\\Gamma$ acts transitively also on the set of non-zero solutions in a big class of composite moduli. Our result is also related to a well-known theorem of Gilman, stating that for any finite non-abelian simple group $G$ and $r\\ge3$, the group $\\mathrm{Aut}\\left(F_{r}\\right)$ acts on at least one $T_{r}$-system of $G$ as the alternating or symmetric group. In this language, our main result translates to that for most primes $p$, the group $\\mathrm{Aut}\\left(F_{2}\\right)$ acts on a particular $T_{2}$-system of $\\mathrm{PSL}\\left(2,p\\right)$ as the alternating or symmetric group.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-02-27T16:29:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11D25",
      "20B15",
      "20B25",
      "20E05"
    ],
    "keywords": [
      "composite moduli",
      "markoff group",
      "transformations",
      "symmetric group",
      "non-zero solutions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}