A Characterization of Higher Rank Symmetric Spaces Via Bounded Cohomology

. 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 is a higher rank symmetric space iff is injective (and otherwise the kernel is infi...

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Bibliographic Details
Published in:Geometric and functional analysis 2009-05, Vol.19 (1), p.11-40
Main Authors: Bestvina, Mladen, Fujiwara, Koji
Format: Article
Language:English
Subjects:
Analysis
Mathematics
Mathematics and Statistics
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Summary:. 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 is a higher rank symmetric space iff 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.
ISSN:1016-443X
1420-8970
DOI:10.1007/s00039-009-0717-8