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  "body": "\n## Definition\n\nLet $G$ be a [[finitely presented group|TJKANTKASZ]] with a finite [[presentation|0JGEQXQ4AC]] $\\langle S \\mid R \\rangle$, and let $F(S)$ be the [[free group|0KTW26C80J]] on $S$.\nLet $w$ be a word over $S \\cup S^{-1}$ that represents the identity of $G$.\nIts *area* $\\mathrm{Area}(w)$ is the least $N \\ge 0$ such that $w$ is a product of $N$ [[conjugates|0DXD4QSZ8A]] of relators and their inverses in $F(S)$.\nThis means an equality $w = \\prod_{i=1}^{N} u_i \\, r_i^{\\pm 1} \\, u_i^{-1}$ in $F(S)$, with words $u_i$ and relators $r_i \\in R$.\nThe *Dehn function* of the presentation is the function $\\delta \\colon \\N \\to \\N$ given by\n$$\\delta(n) = \\max \\set{\\mathrm{Area}(w) : \\abs{w} \\le n, \\ \\bar w = 1 \\text{ in } G}.$$\n\n## Remarks\n\nFor functions $f, g \\colon \\N \\to \\N$, write $f \\preceq g$ if there is $C \\ge 1$ with $f(n) \\le C g(Cn + C) + Cn + C$ for all $n$, and $f \\simeq g$ if $f \\preceq g \\preceq f$.\nThe Dehn functions of two finite presentations of the same group are $\\simeq$-equivalent, so one speaks of the Dehn function of $G$.\n\nExample: for $\\Z^2 = \\langle a, b \\mid aba^{-1}b^{-1} \\rangle$, the word $a^n b^n a^{-n} b^{-n}$ of length $4n$ has area $n^2$, and every null-homotopic word of length $n$ has area at most a constant times $n^2$; so $\\delta(n) \\simeq n^2$.\n\nExample: free groups and finite groups have Dehn function $\\simeq n$.\n\nNon-example: the [[lamplighter group|4GNPFMT7DZ]] is finitely generated but not finitely presented, so it has no Dehn function in this sense.\n\nA finitely presented group has a solvable word problem if and only if its Dehn function is bounded above by a recursive function.\nThe [[Baumslag–Solitar group|429JJ5ZNBY]] $\\langle a, b \\mid b a b^{-1} = a^2 \\rangle$ has an exponential Dehn function.\n",
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