Abstract:
Hyperbolic space is the classical non-Euclidean space introduced by Lobachevksy, Bolyai and Gauss, in which the sum of the angles of a triangle is less than \pi. Hyperbolic n-manifolds are spaces that have the local geometry of hyperbolic n-space. The study of these spaces, besides being natural from the viewpoint of classical geometry, has important connections with differential geometry, algebraic geometry, complex analysis, number theory and topology.
The connection with topology, which turns out to be especially strong in dimension 3, arises in part from the Mostow Rigidity Theorem, which implies that the geometry of compact hyperbolic manifolds of dimension at least 3 is entirely determined by their topology. This means that any geometric invariant of a hyperbolic manifold may be regarded as a topological invariant. The theorem itself says nothing about the question of how such geometric invariants as the volume (which is a positive real number) are related to more classical topological invariants of a 3-manifold. I will describe some progress on this question, which has been a focus of my research for many years. This includes joint work with Phil Wagreich, Marc Culler, Gilbert Baumslag, Ian Agol, and Jason DeBlois, and some very recent work with Rosemary Guzman.
704 Thackeray Hall