{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $M$ be an orientable [[hyperbolic 3-manifold|GHVW1Z35BS]] of finite volume that is the interior of a compact $3$-manifold $\\bar{M}$ whose boundary consists of tori $T_1, \\ldots, T_n$.\nA *slope* on $T_i$ is an isotopy class of essential simple closed curves on $T_i$.\nFor slopes $s_1, \\ldots, s_n$, let $M(s_1, \\ldots, s_n)$ be the closed manifold obtained by [[Dehn filling|ZA405B0Z1S]].\nIt is made by gluing a solid torus to each $T_i$ so that a meridian is glued to $s_i$.\nThen for each $i$ there is a finite set $E_i$ of slopes on $T_i$ such that $M(s_1, \\ldots, s_n)$ is hyperbolic whenever $s_i \\notin E_i$ for every $i$.\n\n## Remarks\n\nThe theorem is due to Thurston.\nThe complete structure on $M$ lies in a space of incomplete hyperbolic structures, parametrised near it by the shapes of ideal tetrahedra.\nNearby incomplete structures whose metric completion is a closed manifold give the filled manifolds.\nThurston also showed that the volumes of the filled manifolds are smaller than the volume of $M$ and tend to it as all slopes go to infinity.\nFor the complement of the [[figure-eight knot|C01XBGAHC6]], exactly ten slopes give a non-hyperbolic filling: $\\infty$, $0$, $\\pm 1$, $\\pm 2$, $\\pm 3$ and $\\pm 4$.\nThe finite set cannot be empty in general: the filling along the meridian of a knot complement gives back $S^3$, which is not hyperbolic.\n\n## Explanations\n\n### L1 human\n\nStart from a hyperbolic space with some tube-shaped ends, and close each end by gluing in a solid doughnut.\nThere are infinitely many ways to glue each doughnut.\nThurston's theorem says that all of them, except for finitely many on each end, give a closed hyperbolic manifold.\n\n### L2 human\n\nThe theorem produces infinitely many closed hyperbolic $3$-manifolds from each cusped one, for example from knot complements.\nSince every closed orientable $3$-manifold is obtained by surgery on a link in $S^3$, it suggests that hyperbolic manifolds are generic.\nThe exceptional slopes are well studied: Lackenby and Meyerhoff proved that a one-cusped manifold has at most ten of them, so the figure-eight knot complement is extremal.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "about": [
      "GHVW1Z35BS",
      "ZA405B0Z1S"
    ],
    "kind": "statement",
    "msc": [
      "57K32",
      "57K30"
    ],
    "name": "Hyperbolic Dehn surgery theorem",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Hyperbolic_Dehn_surgery"
      }
    ],
    "tag": "BB28P2ED5H",
    "uses": [
      "GHVW1Z35BS",
      "ZA405B0Z1S",
      "C01XBGAHC6"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/BB28P2ED5H/",
  "links": [
    {
      "field": "about",
      "target": "GHVW1Z35BS"
    },
    {
      "field": "about",
      "target": "ZA405B0Z1S"
    },
    {
      "field": "uses",
      "target": "GHVW1Z35BS"
    },
    {
      "field": "uses",
      "target": "ZA405B0Z1S"
    },
    {
      "field": "uses",
      "target": "C01XBGAHC6"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/BB28P2ED5H.md",
  "msc": [
    "57K32",
    "57K30"
  ],
  "name": "Hyperbolic Dehn surgery theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/BB2/BB28P2ED5H.md",
  "status": null,
  "tag": "BB28P2ED5H",
  "uri": "m2://BB28P2ED5H",
  "url": "https://math2.tamnd.com/wiki/hyperbolic-dehn-surgery-theorem/",
  "writing": {
    "tag": "BB28P2ED5H",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}