CAT(0) spaces of higher rank

Ballmann's Rank Rigidity Conjecture predicts that a CAT(0) space of higher rank with a geometric group action is rigid -- isometric to a Riemannian symmetric space, a Euclidean building, or splits as a direct product. We confirm this conjecture in rank 2. For CAT(0) spaces of higher rank $n\geq 2$ we prove rigidity if the space contains a periodic $n$-flat and every complete geodesic lies in some $n$-flat. This does not require a geometric group action.