Mathematical Sciences: Solid Tubes in Hyperbolic 3-Manifolds and Applications

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9626561 Meyerhoff Recent work of D. Gabai indicates that certain fundamentally important open problems in 3-manifold theory will be solved if an appropriate understanding of solid tube neighborhoods around short geodesics in hyperbolic 3-manifolds can be developed. For example, if it can be shown that sufficiently large solid tube neighborhoods can be found in any hyperbolic 3-manifold, then it would follow that closed (irreducible) 3-manifolds that are homotopy equivalent to closed hyperbolic 3-manifolds are themselves hyperbolic. This project has as its goal to gain a deep understanding of solid tubes in hyperbolic 3-manifolds and to apply this understanding to fundamental problems in 3-manifold theory. The methods employed involve parametrizing the possible ways that shortest geodesics can sit in hyperbolic 3-manifolds and then analyzing the resulting parameter space by means of rigorous computer programs. This work is joint with D. Gabai and N. Thurston. In the late 18th century, Henri Poincare pioneered the study of 3-manifolds. One of his best methods was to analyze all loops in the 3-manifold in question; in particular, he studied the so-called fundamental group of the 3-manifold. Poincare's general question was, if I understand the fundamental group, do I thereby understand the 3-manifold completely? (The infamous Poincare Conjecture is a specific instance of this general question.) In the late 1970's and early 1980's, William Thurston revolutionized the study of 3-manifolds by showing that most 3-manifolds have hyperbolic structures (hyperbolic geometry is the most important non-Euclidean geometry). This leads to a natural and important Poincare type question: If a compact (irreducible) 3-manifold has the same fundamental group as a compact hyperbolic 3-manifold, must it be a hyperbolic 3-manifold as well? The goal of this project is to answer this question in the affirmative. The method is to reduce the question to an analysis of a certain region in a 6-dimensional space that is related to solid tubes around shortest geodesics in hyperbolic 3-manifolds. The plan is to break this region up into about a billion sub-boxes and to deal with each of these sub-boxes separately. Naturally, this involves the use of a (rigorous!) computer program. ***