The Topology of Hyperbolic 3-Manifolds

University Of Illinois At Chicago, Chicago IL

Investigators

Abstract

Culler and Shalen are continuing their study of the topological structure of hyperbolic 3-manifolds. The context for this work is the on-going unification of the geometric and topological theories of 3-manifolds. The results can be viewed in terms of the theorem, which is a consequence of work of Gromov, Thurston and Jorgensen, that the set of volumes of closed hyperbolic 3-manifolds is a well-ordered set of real numbers. The proposed research aims to understand the topological properties of the 3-manifolds with volume less than a given threshold value. Alternatively, the goal is to determine the volume which corresponds to the ordinal at which a given topological property first appears. The techniques used range from very classical topological methods, such as a refined version of the tower construction used in proving the Loop Theorem, to the most recent developments, including the proof of the Marden Tameness conjecture and Perelman's estimates on the change of volume under Ricci flow. Interesting connections with group theory and combinatorial topology are also involved. The spaces which are being studied in this research project, namely 3-dimensional manifolds, serve as mathematical models of the spatial aspect of a possible physical universe. New connections between modern physics and the mathematical theory of 3-manifolds are being discovered at a rapidly accelerating pace. The mathematical theory has traditionally been divided into topology and geometry, where geometry focuses on quantities which can be measured, such as lengths, angles, areas or volumes, and topology focuses on global properties that are preserved even when the geometric features are distorted. However, these two aspects of the subject are closely related, and there are many examples of results which relate geometric properties and topological properties. Recent mathematical achievements, beginning with the Mostow Rigidity theorem and continuing up to the recent proofs of Marden's Tameness Conjecture and Thurston's Geometrization Conjecture, are leading toward a unification of the geometrical and topological theories of 3-manifolds. The research supported by this grant concentrates on hyperbolic manifolds, a class which includes the vast majority of 3-manifolds with a homogeneous geometric structure, and aims to understand in a quantitative sense how topological complexity depends on the geometrical volume of the manifold.