James Farre (Yale University)

Title: Hyperbolic volume and bounded cohomology.

Abstract:

A natural notion of complexity for a closed manifold M is the smallest number of top dimensional simplices it takes to triangulate M. Gromov showed that a variant of this notion called simplicial volume gives a lower bound for the volume of M with respect to any (normalized) Riemannian metric. The heart of his proof factors through the dual notion of bounded cohomology. I will define bounded cohomology of discrete groups illustrated by some examples coming from computing the volumes of geodesic simplices in hyperbolic space. Although bounded cohomology is often an unwieldy object evading computation, we give some conditions for volume classes to be non-vanishing in low dimensions. We then ask, "When do higher dimensional volume classes vanish?"