{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $F(S)$ be the [[free group|0KTW26C80J]] on a set $S$, with elements written as freely reduced words.\nLet $\\abs{w}$ be the length of a word $w$.\nA set $R$ of cyclically reduced words in $F(S)$ is *symmetrized* if it is closed under taking inverses and cyclic permutations.\nLet $R$ be symmetrized.\nA *piece* relative to $R$ is a nonempty word $u$ such that two distinct elements $r_1 \\ne r_2$ of $R$ both begin with $u$.\nThat is, $r_1 = u v_1$ and $r_2 = u v_2$ with no cancellation.\n\n1. Let $\\lambda$ be a real number with $0 \u003c \\lambda \u003c 1$. Then $R$ satisfies the *condition $C'(\\lambda)$* if $\\abs{u} \u003c \\lambda \\abs{r}$ whenever $r \\in R$ and $r = u v$ with $u$ a piece.\n2. For an integer $p \\ge 2$, $R$ satisfies the *condition $C(p)$* if no element of $R$ is a product of fewer than $p$ pieces.\n\nA [[presentation|0JGEQXQ4AC]] $\\langle S \\mid R_0 \\rangle$ is a *small cancellation presentation* of type $C'(\\lambda)$ or $C(p)$ if the symmetrized closure of $R_0$ satisfies the condition.\n*Small cancellation theory* is the study of groups with such presentations.\n\n## Remarks\n\nThe condition $C'(\\lambda)$ implies $C(p)$ for $p$ with $\\lambda \\le 1/(p-1)$; in particular $C'(1/6)$ implies $C(7)$.\nGreendlinger's lemma: if $R$ satisfies $C'(1/6)$ and a nontrivial reduced word $w$ equals $1$ in $\\langle S \\mid R \\rangle$, then $w$ contains a subword of some $r \\in R$ of length greater than $\\abs{r}/2$.\n\nExample: let $S = \\set{a, b, c, d}$ and $r = a b a^{-1} b^{-1} c d c^{-1} d^{-1}$.\nIn $r$ and its cyclic permutations each two-letter subword occurs once, and no two-letter subword of $r^{-1}$ or of its cyclic permutations occurs in $r$.\nSo all pieces have length $1$, and $1 \u003c \\frac{1}{6} \\cdot 8$, so the presentation satisfies $C'(1/6)$.\n\nNon-example: $\\langle a, b \\mid a b a^{-1} b^{-1} \\rangle$, a presentation of $\\Z^2$, does not satisfy $C'(1/6)$.\nEach letter is a piece, but the relator has length $4$ and $1 \\ge \\frac{1}{6} \\cdot 4$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "3A1D37TVF6",
    "kind": "definition",
    "msc": [
      "20F06",
      "20F05"
    ],
    "name": "Small cancellation theory",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Small_cancellation_theory"
      }
    ],
    "tag": "PW2TVRD47Y",
    "uses": [
      "0KTW26C80J",
      "0JGEQXQ4AC"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/PW2TVRD47Y/",
  "links": [
    {
      "field": "defines",
      "target": "3A1D37TVF6"
    },
    {
      "field": "uses",
      "target": "0KTW26C80J"
    },
    {
      "field": "uses",
      "target": "0JGEQXQ4AC"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/PW2TVRD47Y.md",
  "msc": [
    "20F06",
    "20F05"
  ],
  "name": "Small cancellation theory",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/PW2/PW2TVRD47Y.md",
  "status": null,
  "tag": "PW2TVRD47Y",
  "uri": "m2://PW2TVRD47Y",
  "url": "https://math2.tamnd.com/wiki/small-cancellation-theory-definition/",
  "writing": {
    "tag": "PW2TVRD47Y",
    "kind": "definition",
    "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
      }
    ]
  }
}