Three-dimensional manifolds appeal to us at a very basic level, but they are very difficult to understand. There are many ways to construct examples of 3-manifolds, but the simplest sorts of questions about the constructed manifolds often cannot be answered. For example, it is very difficult to tell whether two construction schemes yield the same or different manifolds. What is needed is a computable collection of invariants which distill the vast amount of information contained in the three-manifold into a useable form. The standard invariants, Euler characteristic, homology, fundamental group, either contain almost no information or are very difficult to work with. Hyperbolic 3-manifolds are the most important and most complicated class of geometric 3-manifolds. The hyperbolic structures on these manifolds can be used to describe such invariants as the Chern-Simons invariant, the volume, and the eta-invariant. By Mostow's theorem, these geometric invariants are actually topological invariants, at least in the closed case. Recent research indicates that these invariants may be very useful in helping us to understand 3-manifolds. This project mounts a multi-sided attack on understanding these invariants.
