{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "PW2TVRD47Y"
    }
  ],
  "body": "\n## Summary\n\nSmall cancellation theory studies groups given by [[presentations|0JGEQXQ4AC]] whose relators have only short common pieces.\nUnder such conditions, such as $C'(1/6)$, a nontrivial word equal to the identity must contain a long part of some relator.\nThis gives Dehn's algorithm for the [[word problem|PBGTTAVMMT]], and finitely presented $C'(1/6)$ groups are [[hyperbolic|JWJ8YPR7EB]].\nThe theory is also a tool to build groups with unusual properties, as in the construction of [[Tarski monster groups|TE19BGFFF1]].\n\n## Explanations\n\n### L1 human\n\nA presentation lists generators and relators, words that are declared equal to $1$.\nIf two relators share a long piece, multiplying them can cancel a lot and produce surprising consequences.\nSmall cancellation conditions forbid this: any piece shared by two relators is a small part of each.\nThen the relators behave almost independently, and one can tell when a word equals $1$ by a simple greedy rule.\nFor example, the fundamental group of a closed orientable surface of genus $2$, $\\langle a, b, c, d \\mid [a, b][c, d] \\rangle$, satisfies the condition $C'(1/6)$.\n",
  "commit": "efe85cee8dbc57d85bedbacfb97b99d1b7bee843",
  "front": {
    "kind": "concept",
    "msc": [
      "20F06",
      "20F05"
    ],
    "name": "Small cancellation theory",
    "same_as": {
      "wikidata": "Q7543015",
      "wikipedia": "Small cancellation theory"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Small_cancellation_theory"
      }
    ],
    "tag": "3A1D37TVF6"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/3A1D37TVF6/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/3A1D37TVF6.md",
  "msc": [
    "20F06",
    "20F05"
  ],
  "name": "Small cancellation theory",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/3A1/3A1D37TVF6.md",
  "status": null,
  "tag": "3A1D37TVF6",
  "uri": "m2://3A1D37TVF6",
  "url": "https://math2.tamnd.com/wiki/small-cancellation-theory/",
  "writing": {
    "tag": "3A1D37TVF6",
    "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
      }
    ]
  }
}