{
  "id": "1505.07886",
  "version": "v1",
  "published": "2015-05-28T23:35:30.000Z",
  "updated": "2015-05-28T23:35:30.000Z",
  "title": "Profinite rigidity, fibering, and the figure-eight knot",
  "authors": [
    "Martin R Bridson",
    "Alan W Reid"
  ],
  "comment": "15 pages, no figures",
  "categories": [
    "math.GT",
    "math.GR"
  ],
  "abstract": "We establish results concerning the profinite completions of 3-manifold groups. In particular, we prove that the complement of the figure-eight knot $S^3-K$ is distinguished from all other compact 3-manifolds by the set of finite quotients of its fundamental group. In addition, we show that if $M$ is a compact 3-manifold with $b_1(M)=1$, and $\\pi_1(M)$ has the same finite quotients as a free-by-cyclic group $F_r\\rtimes\\mathbb{Z}$, then $M$ has non-empty boundary, fibres over the circle with compact fibre, and $\\pi_1(M)\\cong F_r\\rtimes_\\psi\\mathbb{Z}$ for some $\\psi\\in{\\rm{Out}}(F_r)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-05-28T23:35:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20E18",
      "57M25",
      "20E26"
    ],
    "keywords": [
      "figure-eight knot",
      "profinite rigidity",
      "finite quotients",
      "profinite completions",
      "free-by-cyclic group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150507886B"
    }
  }
}