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  "body": "\n## Summary\n\nA random group is a group given by a [[presentation|0JGEQXQ4AC]] whose relators are chosen at random.\nThe random model has a size parameter that tends to infinity.\nOne then asks which properties hold with probability tending to $1$.\nGromov's density model has a sharp threshold at density $\\frac{1}{2}$.\nBelow it, random groups are infinite and [[hyperbolic|JWJ8YPR7EB]]; above it, they are trivial or of order $2$.\nRandom groups show what a typical finitely presented group looks like, and they provide examples of groups with unusual combinations of properties.\n\n## Explanations\n\n### L1 human\n\nChoose a few letters as generators and write down many random words in them as relations.\nThe group you get is a random group.\nWith few relations compared to the number of possible words, the group is almost surely infinite and looks like a tree at large scales.\nWith too many relations, they collapse everything, and the group is almost surely trivial or has two elements.\n",
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