Geometry and Topology of Arithmetic 3-manifolds.

Saint Louis University, Saint Louis MO

Investigators

Abstract

Proposal: DMS-0072515 PI: Anneke Bart and Kevin Scannell Abstract The proposers are interested in geometric and topological properties of hyperbolic 3-manifolds arising from the Bianchi groups, and more generally in the properties of arithmetically-defined 3-manifolds. The proposers intend to unify two of the main themes of research in this area, namely, the cuspidal cohomology problem with non-trivial coefficients, and the enumeration and description of immersed totally geodesic surfaces. The proposers intend to explore possible connections between non-vanishing results in cuspidal cohomology and generalized bending deformations supported on totally geodesic surfaces. The Bianchi groups are particularly interesting mathematical objects as they lie at the crossroads of several distinct area of pure mathematics: number theory, infinite group theory, geometry, and topology. They have been studies from these diverse points of view for more than one hundred years. Certain three-dimensional spaces arise in a natural way from the Bianchi groups; these spaces come equipped with a "hyperbolic geometry". This, in a nutshell, means that pairs of straight lines generically diverge from one another unlike Euclidean geometry. Three-dimensional spaces with a hyperbolic geometry are central to the study of general three-dimensional spaces as they represent, in a certain sense, the "generic" examples. Hyperbolic spaces are among the most difficult to understand, and therefore the proposers aim to elucidate the geometry of these important examples mentioned above.