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  "body": "\n## Definition\n\nLet $X$ be a [[CAT(0) space|1FAF2VNA4G]], that is, a geodesic metric space with the following property.\nFor every geodesic triangle $\\Delta$, let $\\bar\\Delta$ be a triangle in $\\R^2$ with the same side lengths.\nThen $d(p, q) \\le \\norm{\\bar p - \\bar q}$ for all points $p, q$ of $\\Delta$ and their comparison points $\\bar p, \\bar q$ in $\\bar\\Delta$.\nLet a group $G$ [[act|0WCY42EV9V]] on $X$ by [[isometries|02HPP5VCZH]].\n\n1. The action is *proper* if each $x \\in X$ has a radius $r \u003e 0$ such that $\\set{g \\in G : g B(x, r) \\cap B(x, r) \\neq \\emptyset}$ is finite.\n2. The action is *cocompact* if there is a compact set $K \\subseteq X$ with $GK = X$.\n\nThe group $G$ is a *CAT(0) group* if it admits a proper and cocompact action by isometries on some CAT(0) space.\n\n## Remarks\n\nExample: $\\Z^n$ acts on $\\R^n$ by translations.\nThe Euclidean space $\\R^n$ is CAT(0), since its triangles are exactly as thin as their comparison triangles.\nThe action is proper, because a ball of radius $1/3$ meets only its own translate, and cocompact, with $K = [0, 1]^n$.\n\nExample: a free group of finite rank acts properly and cocompactly on its Cayley graph for a free basis, which is a tree, and trees are CAT(0).\n\nNon-example: the integer Heisenberg group is not a CAT(0) group.\nIn a CAT(0) group every solvable subgroup is virtually abelian, and the Heisenberg group is nilpotent but not virtually abelian.\n\nSuch an action is also called geometric.\nEvery CAT(0) group is finitely presented, and its [[Dehn function|Q4HGMRA9JQ]] is at most quadratic.\n",
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