{
  "id": "1104.3062",
  "version": "v1",
  "published": "2011-04-15T14:06:10.000Z",
  "updated": "2011-04-15T14:06:10.000Z",
  "title": "On malnormal peripheral subgroups in fundamental groups of 3-manifolds",
  "authors": [
    "Pierre de la Harpe",
    "Claude Weber"
  ],
  "categories": [
    "math.GR"
  ],
  "abstract": "Let $K$ be a non-trivial knot in the 3-sphere, $E_K$ its exterior, $G_K = \\pi_1(E_K)$ its group, and $P_K = \\pi_1(\\partial E_K) \\subset G_K$ its peripheral subgroup. We show that $P_K$ is malnormal in $G_K$, namely that $gP_Kg^{-1} \\cap P_K = \\{e\\}$ for any $g \\in G_K$ with $g \\notin P_K$, unless $K$ is in one of the following three classes: torus knots, cable knots, and composite knots; these are exactly the classes for which there exist annuli in $E_K$ attached to $T_K$ which are not boundary parallel (Theorem 1 and Corollary 2). More generally, we characterise malnormal peripheral subgroups in the fundamental group of a compact orientable irreducible 3-manifold with boundary a non-empty union of tori (Theorem 3). Proofs are written with non-expert readers in mind. Half of our paper (Sections 7 to 10) is a reminder of some three-manifold topology as it flourished before the Thurston revolution. In a companion paper [HaWeOs], we collect general facts on malnormal subgroups and Frobenius groups, and we review a number of examples.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-04-15T14:06:10.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fundamental group",
      "characterise malnormal peripheral subgroups",
      "collect general facts",
      "boundary parallel",
      "frobenius groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1104.3062D"
    }
  }
}