9626578 Canary Professor Canary will explore a variety of conjectures concerning hyperbolic 3-manifolds and the deformation theory of Kleinian groups. Several of these conjectures are motivated by Marden's tameness conjecture, which predicts that every hyperbolic 3-manifold with finitely generated fundamental group is topologically tame, i.e., homeomorphic to the interior of a compact 3-manifold. Professor Canary has previously established that topological tameness has strong consequences for the geometry of a hyperbolic 3-manifold. He will also study the relationship between the algebraic limit and the geometric limit of a sequence of Kleinian groups. This study is inspired by Thurston's ending lamination conjecture, which provides a conjectural classification of all hyperbolic 3-manifolds. A 3-manifold is a mathematical space such that about any point there is a neighborhood which can be identified with a ball in 3-dimensional space. One can imagine building 3-dimensional manifolds by gluing together 3-dimensional blocks. Of course, this gluing would have to be an abstract gluing in general, not one which could be done in 3-dimensional space. For example, consider the 3-manifold obtained by taking the unit cube and gluing the top to the bottom, the front to the back, and the left side to the right side. A Riemannian metric is a way of measuring distances and angles in a 3-manifold. For instance, the world we live in is a 3-manifold with a Riemannian metric. In the 1970's, Wiliam Thurston conjectured that every 3-manifold can be cut up, in a canonical way, so that each piece has a Riemannian metric of one of 8 geometric types. He proved his conjecture for large classes of three-manifolds. The 3-manifolds which admit seven of the eight geometric types of Riemannian metrics are completely classified and well-understood. 3-manifolds that admit the eighth type of metric are called hyperbolic and are currently a subject of intense interest and act ivity in the fields of geometry and topology. Professor Canary is investigating the relationship between the topology and the geometry of hyperbolic 3-manifolds. ***
