CAT(0) spaces of higher rank

Abstract

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 ≥ 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.