One of the central goals in the geometric theory of 3-manifolds is understand the connection between the topological, combinatorial, and geometric properties of 3-manifolds. It is important to have explicit, quantitative information about a geometric structure and to understand how it is connected with the combinatorial and topological properties of the manifold. The model manifolds constructed by Minsky and used in the solution of the Ending Lamination Conjecture in Kleinian groups are determined by combinatorial data and reflect geometric properties of the mapping class group. These models should be useful in studying closed hyperbolic 3-manifolds. On the other hand, techniques that have been successful in studying closed manifolds, like hyperbolic Dehn surgery, deformation theory, and ideal triangulations should help provide further refinements in our understanding of Kleinian groups. The goal of this project is to combine the analytic and geometric techniques developed over the last several years by the PI and his collaborators with some of the new ideas that have emerged during the recent solutions of major conjectures in Kleinian groups.
The study of 3-dimensional manifolds combines geometric, algebraic, and analytic tools. It can be a very approachable subject because the objects of study include things like knots that are easy to describe and visualize geometrically. Even the deeper properties, like hyperbolic structures, can be presented in an inviting manner with current graphical capabilities. On the other hand, understanding knots and 3-manifolds can have significant consequencesin other areas of mathematics and even in physics and biochemistry.
