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  "body": "\n## Summary\n\nA CAT(0) group is a group that acts properly and cocompactly by isometries on a [[CAT(0) space|1FAF2VNA4G]].\nSuch a space is a geodesic space whose triangles are at least as thin as Euclidean triangles.\nIt is the group-theoretic form of nonpositive curvature.\nExamples include $\\Z^n$, free groups, fundamental groups of closed nonpositively curved Riemannian manifolds, [[right-angled Artin groups|6D1DV09D6P]] and [[Coxeter groups|G6FND7X69J]].\nCAT(0) groups are finitely presented, have a solvable word problem, and have a [[Dehn function|Q4HGMRA9JQ]] at most quadratic.\n\n## Explanations\n\n### L1 human\n\nDraw a triangle on a saddle surface: its sides bend inwards compared with a flat triangle of the same side lengths.\nA CAT(0) space is a space where every triangle is at least this thin, never fatter than a flat one.\nA CAT(0) group is a group of symmetries of such a space.\nIt moves points around in a discrete way, and the translates of one compact piece cover the whole space.\nFor example, $\\Z^2$ acts on the flat plane by translations, and the unit square tiles the plane.\n",
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