an:05575931
Zbl 1203.53041
Bestvina, Mladen; Fujiwara, Koji
A characterization of higher rank symmetric spaces via bounded cohomology
EN
Geom. Funct. Anal. 19, No. 1, 11-40 (2009).
1016-443X 1420-8970
2009
j
53C24 20F65
bounded cohomology; quasi-homomorphisms, higher rank symmetric spaces; Rank Rigidity theorem; rank 1 isometries
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.