{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $\\mathrm{PSL}_2(\\C) = \\mathrm{SL}_2(\\C) / \\set{\\pm I}$, where [[SL(2,C)|J58ED1GV6N]] has the topology of a subset of $\\C^4$ and $\\mathrm{PSL}_2(\\C)$ has the quotient topology.\nA *Kleinian group* is a [[discrete subgroup|5M7D1BSHVZ]] of $\\mathrm{PSL}_2(\\C)$.\n\n## Remarks\n\nThe group $\\mathrm{PSL}_2(\\C)$ acts on the [[Riemann sphere|0J78KBPFZB]] by [[Möbius transformations|0SG74WBFA2]] and on hyperbolic $3$-space $\\mathbb H^3$ by isometries.\nA subgroup is Kleinian if and only if its action on $\\mathbb H^3$ is properly discontinuous.\nSome authors also require that the group act properly discontinuously on a nonempty open subset of the Riemann sphere.\n\nExample: the Picard group $\\mathrm{PSL}_2(\\Z[i])$, of matrices with Gaussian integer entries, is Kleinian.\nDistinct Gaussian integers are at distance at least $1$, so a small neighbourhood of $\\pm I$ contains no other element of the group.\n\nNon-example: the group $\\mathrm{PSL}_2(\\Z[\\sqrt2])$ is not Kleinian.\nIt contains the translations $z \\mapsto z + a + b\\sqrt2$ with $a, b \\in \\Z$, and their translation lengths come arbitrarily close to $0$ without being $0$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "27QRARKWD5",
    "kind": "definition",
    "msc": [
      "30F40",
      "57K32",
      "20H10"
    ],
    "name": "Kleinian group",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Kleinian_group"
      }
    ],
    "tag": "MC4EFHWRJG",
    "uses": [
      "5M7D1BSHVZ",
      "J58ED1GV6N",
      "0SG74WBFA2",
      "0J78KBPFZB"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/MC4EFHWRJG/",
  "links": [
    {
      "field": "defines",
      "target": "27QRARKWD5"
    },
    {
      "field": "uses",
      "target": "5M7D1BSHVZ"
    },
    {
      "field": "uses",
      "target": "J58ED1GV6N"
    },
    {
      "field": "uses",
      "target": "0SG74WBFA2"
    },
    {
      "field": "uses",
      "target": "0J78KBPFZB"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/MC4EFHWRJG.md",
  "msc": [
    "30F40",
    "57K32",
    "20H10"
  ],
  "name": "Kleinian group",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/MC4/MC4EFHWRJG.md",
  "status": null,
  "tag": "MC4EFHWRJG",
  "uri": "m2://MC4EFHWRJG",
  "url": "https://tamnd.github.io/math2/wiki/kleinian-group-definition/",
  "writing": {
    "tag": "MC4EFHWRJG",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}