The Topology and Geometry of Hyperbolic 3-Manifolds

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

The subject of geometry and deformation theory of hyperbolic 3-manifolds has seen considerable change and advancement over the last three years. Marden's Tameness Conjecture, Thurston's Ending Lamination Conjecture, the Bers-Sullivan-Thurston Density Conjecture and Ahlfors' Measure Conjecture have all been established. Moreover, the deformation theory of hyperbolic 3-manifolds has been revealed to be more complicated than previously expected. Powerful new techniques have been introduced to the field, including model manifolds, the geometry of the curve complex, drilling theorems, and end reductions. Prof. Canary proposes to use the new techniques to improve our understanding of the geometry of hyperbolic 3-manifolds, to further illuminate the still mysterious deformation theory of hyperbolic 3-manifolds and to explore new directions for research in the field. Prof. Canary proposes to continue his study of the topology and geometry of 3-dimensional hyperbolic manifolds. A 2-dimensional manifold or surface is a space which looks locally like 2-dimensional Euclidean space; examples are given by the surfaces of familiar 3-dimensional objects such as footballs or doughnuts. Similarly, a 3-manifold is a space that looks locally like 3-dimensional Euclidean space. A Riemannian metric is a way of measuring distances and angles in a manifold. For example, the universe we live in is a 3-manifold with a Riemannian metric. For this reason, among others, the study of 3-dimensional manifolds and their geometries is natural and important. It is conjectured that 3-dimensional manifolds can be canonically decomposed into pieces of one of 8 geometric types. The hyperbolic manifolds are the most common and least understood class of such geometric manifolds. In recent years, much progress has been made in understanding such manifolds and Prof. Canary proposes to build on and deepen that understanding.