In this project Bromberg will investigate a number of different questions in 2 and 3 dimensional hyperbolic geometry. One problem is to understand the topology of deformation spaces of hyperbolic 3- manifolds. For example, when are these spaces locally connected? The PI is also working to understand the most general conditions for a sequence of Kleinian groups to converge. Another topic is to find a quantitative version of Thurston's Bounded Image Theorem. The PI is also studying the Mapping Class Group, the group of symmetries of a two dimensional topological surface. One particular question is the asymptotic dimension of this group, which is an important invariant defined by Gromov.
Two and three dimensional manifolds are some of the most important topics mathematicians study. They are the objects we live on and in. Going back to Poincare and Klein mathematicians have used geometric methods to study these topological objects. Thurston's work from the seventies showed the special importance of hyperbolic geometry. More recently many of Thurston's original conjectures have been proven by bringing new techniques into the field and at same time new questions and conjectures. This project will explore these new approaches with the hope of expanding our knowledge of two and three dimensional hyperbolic manifolds.
