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  "body": "\n## Summary\n\nThe Dehn function of a [[finitely presented group|TJKANTKASZ]] measures how many relators are needed to prove that a word of length $n$ equals the identity.\nUp to a standard equivalence it does not depend on the presentation, and it is a [[quasi-isometry|6FKYSH729T]] invariant.\nIt is a geometric measure of the difficulty of the word problem.\nThe word problem is solvable exactly when the Dehn function is bounded by a computable function.\nA group is [[hyperbolic|JWJ8YPR7EB]] if and only if its Dehn function is linear.\n\n## Explanations\n\n### L1 human\n\nA word in the generators that equals the identity can be reduced to the empty word by inserting relators, one at a time, and cancelling.\nThe area of the word is the least number of relators needed.\nThe Dehn function gives the largest area of such a word of length at most $n$.\nIn $\\Z^2 = \\langle a, b \\mid aba^{-1}b^{-1} \\rangle$, the word $a^n b^n a^{-n} b^{-n}$ traces the boundary of an $n \\times n$ square and needs $n^2$ relators, one per unit square.\nSo the Dehn function of $\\Z^2$ grows like $n^2$, like the area enclosed by a loop in the plane.\n",
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