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  "body": "\n## Definition\n\nFix an integer $m \\ge 2$ and a real number $d$ with $0 \\le d \\le 1$, the *density*.\nLet $F_m$ be the [[free group|0KTW26C80J]] on $a_1, \\dots, a_m$.\nFor each integer $\\ell \\ge 1$, let $C_\\ell$ be the [[finite set|0BBTH3YSZA]] of cyclically reduced words of length $\\ell$ in $F_m$.\nLet $R$ be a subset of $C_\\ell$ with $\\floor{(2m - 1)^{d \\ell}}$ elements, chosen uniformly at random among all such subsets.\nA *random group at density $d$ and length $\\ell$* is the group\n$$G = \\langle a_1, \\dots, a_m \\mid R \\rangle.$$\nA property holds *for random groups at density $d$* if the probability that $G$ has it tends to $1$ as $\\ell \\to \\infty$.\n\n## Remarks\n\nThis is Gromov's density model.\nThe number of cyclically reduced words of length $\\ell$ grows like $(2m - 1)^\\ell$, so the density $d$ measures the number of relators on a logarithmic scale.\nGromov's theorem: for $d \u003c \\frac{1}{2}$, random groups at density $d$ are infinite, torsion-free and [[hyperbolic|JWJ8YPR7EB]]; for $d \u003e \\frac{1}{2}$, random groups at density $d$ are trivial or of order $2$.\n\nExample: at density $d = 0$, the set $R$ has exactly one element, so the model produces a random one-relator group.\n\nNon-example: a single fixed group, such as $\\Z^2 = \\langle a, b \\mid aba^{-1}b^{-1} \\rangle$, is not \"random\" in itself.\nThe definition describes a sequence of probability distributions on presentations, and only properties that hold with probability tending to $1$ are meaningful.\n\nOther models fix the number of relators and let their length grow (the few-relator model), or use relators of length $3$ (the triangular model).\n",
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