{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "P52TCSHP7R"
    }
  ],
  "body": "\n## Summary\n\nA Zassenhaus neighbourhood of a [[Lie group|7JBRGCVJRB]] $G$ is a neighbourhood $U$ of the identity with the following property.\nThe elements of any [[discrete subgroup|5M7D1BSHVZ]] that lie in $U$ generate a [[nilpotent group|0Q7MMQHC1P]].\nA theorem of Zassenhaus says that every Lie group has such a neighbourhood.\nIt is the algebraic heart of the [[Margulis lemma|P6WSG7DT4Q]] and of the [[Kazhdan–Margulis theorem|C34RZMKKA4]].\nThe reason is that commutators of elements near the identity are much closer to the identity than the elements themselves.\n\n## Explanations\n\n### L1 human\n\nIn a Lie group, two elements close to the identity almost commute: their commutator is closer still to the identity.\nIn a discrete subgroup, elements cannot get arbitrarily close to the identity.\nSo taking commutators of small elements again and again must end at the identity after finitely many steps.\nThat is what being nilpotent means, and a neighbourhood where this argument works is a Zassenhaus neighbourhood.\nIn an abelian group like $\\R^n$, every neighbourhood works, since all subgroups are abelian.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "22E40",
      "22E15"
    ],
    "name": "Zassenhaus neighbourhood",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Margulis_lemma"
      }
    ],
    "tag": "FZ798EZ6M0"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/FZ798EZ6M0/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/FZ798EZ6M0.md",
  "msc": [
    "22E40",
    "22E15"
  ],
  "name": "Zassenhaus neighbourhood",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/FZ7/FZ798EZ6M0.md",
  "status": null,
  "tag": "FZ798EZ6M0",
  "uri": "m2://FZ798EZ6M0",
  "url": "https://tamnd.github.io/math2/wiki/zassenhaus-neighbourhood/",
  "writing": {
    "tag": "FZ798EZ6M0",
    "kind": "concept",
    "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
      }
    ]
  }
}