A Characterization of Higher Rank Symmetric Spaces Via Bounded Cohomology

Abstract.

Let M be complete nonpositively curved Riemannian manifold of finite volume whose fundamental group Γ does not contain a finite index subgroup which is a product of infinite groups. We show that the universal cover $\tilde{M}$ is a higher rank symmetric space iff $H^{2}_{b} (M; {\mathbb{R}}) \rightarrow H^{2}(M;{\mathbb{R}})$ 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.