{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet *w* ∈ *F*(*A*) be such that *w* = 1 in *G*. Then the *area* of *w*, denoted Area(*w*), is defined as the minimum of the areas of all Van Kampen diagrams with boundary labels *w* (Van Kampen's lemma says that at least one such diagram exists).\n\nOne can show that the area of *w* can be equivalently defined as the smallest *n*≥0 such that there exists a representation (♠) expressing *w* as a product in *F*(*A*) of *n* conjugates of the defining relators.\n",
  "commit": "a799a9852bcfeae9dc83faa3bef60208bb53686b",
  "front": {
    "inventory": [
      "8D310DF0ZJ"
    ],
    "kind": "statement",
    "name": "Area of a word representing the identity",
    "role": "proposition",
    "sources": [
      {
        "locator": "Van Kampen diagram, section Dehn functions and isoperimetric functions \u003e Area of a word representing the identity",
        "mention": "wikipedia://en/18871460#B4W3",
        "note": "adopted",
        "sections": [
          "Statement"
        ],
        "wikipedia": "en:18871460@r1368818220"
      }
    ],
    "tag": "7RQ48RV25W"
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/7RQ48RV25W/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/7RQ48RV25W.md",
  "msc": null,
  "name": "Area of a word representing the identity",
  "prov": null,
  "role": "proposition",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/7RQ/7RQ48RV25W.md",
  "status": null,
  "tag": "7RQ48RV25W",
  "uri": "m2://7RQ48RV25W",
  "url": "https://math2.tamnd.com/wiki/area-of-a-word-representing-the-identity/",
  "writing": {
    "tag": "7RQ48RV25W",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "edge uses out",
        "note": "each definition that the statement needs",
        "have": 0,
        "want": 1
      }
    ]
  }
}
