A CHARACTERIZATION OF HIGHER RANK SYMMETRIC SPACES VIA BOUNDED COHOMOLOGY

Let M be complete nonpositively curved Riemannian manifold of finite volume whose fundamental group Gamma does not contain a finite index subgroup which is a product of infinite groups. We show that the universal cover (M) over tilde is a higher rank symmetric space iff H(b)(2) (M; R) -> H(2)(M; R) is injective (and otherwise the kernel is infinite dimensional). This is the converse of a theorem of Burger-Monod. The proof uses the celebrated Rank Rigidity Theorem, as well as a new construction of quasi-homomorphisms on groups that act on CAT(0) spaces and contain rank 1 elements.