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A characterization of higher rank symmetric spaces via bounded cohomology. (English) Zbl 1203.53041
Summary: 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 \(\widetilde{M}\) 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 \(\mathrm{CAT}(0)\) spaces and contain rank 1 elements.

53C24 Rigidity results
20F65 Geometric group theory
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