Generalizing a construction of quasi-homomorphisms on a group
acting on hyperbolic spaces, we construct quasi-homomorphisms
on a group acting on a CAT(0)-space with rank-1 isometries.
As an application, we characterize finite volume, locally symmetric spaces
of rank at least two among finite volume complete Riemannian manifolds
of non-positive
curvature using quasi-homomorphisms on the fundamental groups.
The construction also applies to the action of a mapping class group
on the Teichmuller space with Weil-Petersson metric.
We show pseudo-Anosov elements are rank-1 in our sense.
This is a joint work with Bestvina.

Lecture #12575

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