Mathematical Sciences: RUI: Cusp Volumes in Hyperbolic 3-Manifolds

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Adams will continue the study of cusp volumes in hyperbolic 3-manifolds. It has been conjectured by W. Thurston that all 3-manifolds can be decomposed into components, each of which has one of eight geometries. He has already proven this conjecture for large classes of 3- manifolds. A complete classification of the 3-manifolds with seven of those geometries has already been completed. The largest and richest case corresponds to those manifolds with the eighth geometry, namely hyperbolic geometry. A hyperbolic 3-manifold has a well-defined volume, which is an invariant for the topology of the manifold. The set of volumes of all finite volume hyperbolic 3-manifolds is a well-ordered subset of the reals. Hence it is of interest to determine some of the hyperbolic manifolds of low volume, a problem suggested by Thurston. A noncompact hyperbolic 3-manifold has additional invariants called cusp volumes. They offer a means of determining manifolds of low volume. This idea has already been utilized by the investigator to determine the smallest hyperbolic manifolds of one or two cusps. The project will continue investigations along these lines both by extending current computer programs in order to create more examples for study and also by utilizing the examples to point the way to proofs that certain manifolds are the smallest manifolds in their classes. In particular, a goal of the project is to prove that the figure-eight knot complement and its sister manifold are the noncompact orientable hyperbolic 3-manifolds of minimal volume, also a conjecture of Thurston's.