CAT(0) group

Summary

A CAT(0) group is a group that acts properly and cocompactly by isometries on a CAT(0) space. Such a space is a geodesic space whose triangles are at least as thin as Euclidean triangles. It is the group-theoretic form of nonpositive curvature. Examples include $\Z^n$, free groups, fundamental groups of closed nonpositively curved Riemannian manifolds, right-angled Artin groups and Coxeter groups. CAT(0) groups are finitely presented, have a solvable word problem, and have a Dehn function at most quadratic.