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  "body": "\n## Summary\n\nA hyperbolic group is a [[finitely generated group|082DWWXMRJ]] whose [[Cayley graph|0GYQK9NDJF]] looks like a tree at large scale: its geodesic triangles are uniformly thin.\nThe notion was introduced by Gromov.\nIt covers free groups, fundamental groups of closed hyperbolic manifolds, and, in a precise sense, most finitely presented groups.\nHyperbolic groups are finitely presented, have a solvable word problem and conjugacy problem, and have a linear [[Dehn function|Q4HGMRA9JQ]].\n\n## Explanations\n\n### L1 human\n\nIn a tree, the three sides of a triangle meet in a tripod, so each side runs right along the other two.\nIn the hyperbolic plane the same is nearly true: triangles are thin, with every side close to the union of the other two.\nA group is hyperbolic if its Cayley graph has this thin-triangle property, with a fixed bound on the thickness.\nThe free group on two generators is the simplest infinite example, since its Cayley graph is a tree.\nThe grid $\\Z^2$ is not hyperbolic: its large triangles are fat, like triangles in the flat plane.\n",
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