{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Summary\n\nGeometric group theory studies [[finitely generated groups|082DWWXMRJ]] through the geometry of spaces on which they act, such as their [[Cayley graphs|0GYQK9NDJF]] with the [[word metric|VCHJVR9NGH]].\nIts basic principle is to study groups up to [[quasi-isometry|6FKYSH729T]], which ignores small-scale details and distortions by bounded factors.\nThe [[Švarc–Milnor lemma|NTK1TVA50A]] links this view to group actions on metric spaces.\nCentral results include [[Gromov's theorem on groups of polynomial growth|RSBFHHPTWX]] and [[Stallings's theorem on ends of groups|GWZN3EKEWD]].\nCentral classes include [[hyperbolic groups|JWJ8YPR7EB]] and [[amenable groups|CEAT9BESKK]].\nThe field grew from combinatorial group theory and low-dimensional topology.\nIt took its modern form in the 1980s and 1990s, largely through the work of Gromov.\n\n## Explanations\n\n### L1 human\n\nA group given by generators can be drawn as a graph: one dot for each element, and a line from $g$ to $gs$ for each generator $s$.\nSeen from far away, this graph has a shape.\nThe Cayley graph of $\\Z^2$ with its two standard generators looks like a grid and, from far away, like a plane.\nThe Cayley graph of a free group on two generators is an infinite tree, which looks nothing like a plane.\nGeometric group theory reads algebraic properties of the group from such large-scale shapes, and the other way around.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "20F65"
    ],
    "name": "Geometric group theory",
    "same_as": {
      "wikidata": "Q2662474",
      "wikipedia": "Geometric group theory"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Geometric_group_theory"
      }
    ],
    "tag": "2T3218RXTD"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/2T3218RXTD/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/2T3218RXTD.md",
  "msc": [
    "20F65"
  ],
  "name": "Geometric group theory",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/2T3/2T3218RXTD.md",
  "status": null,
  "tag": "2T3218RXTD",
  "uri": "m2://2T3218RXTD",
  "url": "https://math2.tamnd.com/wiki/geometric-group-theory/",
  "writing": {
    "tag": "2T3218RXTD",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}