A characterization of higher rank symmetric spaces via bounded cohomology

2007/02/09 by Bestvina, Mladen, Fujiwara, Koji · 4 citations
#FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR)

paper · doi:10.48550/arxiv.math/0702274

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 M is a higher rank symmetric space iff H²_b(M;\R)→ H²(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.

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