[Paper Review] A characterization of higher rank symmetric spaces via bounded cohomology

Mladen Bestvina, Koji Fujiwara|ArXiv.org|Feb 9, 2007

Geometric and Algebraic Topology12 references7 citations

TL;DR

This paper establishes a characterization of higher rank symmetric spaces in terms of bounded cohomology: a complete, finite-volume, nonpositively curved manifold has a higher rank symmetric space as its universal cover if and only if the kernel of the map from bounded to ordinary second cohomology vanishes. The proof uses a new construction of infinite-dimensional quasi-homomorphisms on groups acting on CAT(0) spaces with rank 1 isometries, leveraging the Rank Rigidity Theorem and the WPD condition.

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 $ ilde M$ is a higher rank symmetric space iff $H^2_b(M;\R) o H^2(M;\R)$ 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.

Motivation & Objective

• To establish a converse to the Burger-Monod theorem on bounded cohomology vanishing for irreducible lattices in semisimple Lie groups.
• To characterize higher rank symmetric spaces among finite-volume, nonpositively curved manifolds using bounded cohomology.
• To develop a new method for constructing infinite-dimensional spaces of quasi-homomorphisms on groups acting on CAT(0) spaces with rank 1 isometries.
• To apply the Rank Rigidity Theorem to classify the geometric structure of the universal cover based on cohomological properties.
• To show that the vanishing of the kernel in bounded cohomology corresponds precisely to the universal cover being a higher rank symmetric space.

Proposed method

• Define the space of homogeneous quasi-homomorphisms modulo homomorphisms and bounded functions, identifying it with the kernel of the map from bounded to ordinary cohomology.
• Use the WPD (weak proper discontinuity) condition on group actions on CAT(0) spaces to ensure the existence of rank 1 isometries.
• Construct infinite-dimensional quasi-homomorphisms on groups with rank 1 isometries using ruled triangles and curvature bounds in Teichmüller space with the Weil-Petersson metric.
• Apply the Gauss-Bonnet theorem and curvature estimates to bound areas and lengths in neighborhoods of geodesic axes, proving boundedness of projections.
• Leverage the Rank Rigidity Theorem to eliminate reducible or Euclidean factors in the universal cover, reducing to symmetric spaces or rank 1 isometries.
• Use the fact that stable commutator length vanishes if and only if the kernel of bounded cohomology map vanishes, linking to Bavard's theorem.

Experimental results

Research questions

• RQ1 Can the vanishing of the kernel in the bounded cohomology map characterize higher rank symmetric spaces among finite-volume, nonpositively curved manifolds?
• RQ2 Under what group-theoretic and geometric conditions does the space of quasi-homomorphisms remain infinite-dimensional?
• RQ3 How can bounded cohomology be used to distinguish symmetric spaces of higher rank from other nonpositively curved manifolds?
• RQ4 What role do rank 1 isometries and the WPD condition play in the structure of the quasi-homomorphism space?
• RQ5 To what extent can the Burger-Monod vanishing result be reversed using geometric group theory tools?

Key findings

• The universal cover of a finite-volume, nonpositively curved manifold is a higher rank symmetric space if and only if the kernel of the map from bounded to ordinary second cohomology is trivial.
• If the fundamental group contains no finite-index product of infinite groups and no virtually cyclic subgroup, then the kernel is trivial precisely when the universal cover is a higher rank symmetric space.
• For groups acting on CAT(0) spaces with a rank 1 isometry and satisfying the WPD condition, the space of quasi-homomorphisms is infinite-dimensional.
• The Teichmüller space with the Weil-Petersson metric supports an action of the mapping class group that satisfies WPD, enabling the application of the main theorem.
• The projection of any ball disjoint from a geodesic axis in Teichmüller space has uniformly bounded diameter, a key technical result for WPD verification.
• The curvature bounds in neighborhoods of geodesic axes in Teichmüller space lead to area and length estimates that imply finite-diameter projections, supporting the WPD condition.