Deformations of hyperbolic 3-manifolds

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

Investigators

Abstract

Proposal: DMS-0072062 Abstract: Professor Bromberg plans to study spaces of hyperbolic metrics on a fixed 3-manifold. Two types of questions will be investigated. The first of these is a study of hyperbolic cone-manifolds. Finite volume cone-manifolds arise naturally as a link between cusped hyperbolic 3-manifolds and closed hyperbolic 3-manifolds. Bromberg's goal is to parameterize spaces of infinite volume hyperbolic cone-manifolds and use this parameterization to understand non-singular hyperbolic structures. The second question involves the topology of deformation spaces of complete, infinite volume metrics on open 3-manifolds. Anderson and Canary have shown that the space of such metrics on a manifold of fixed homotopy type has a very complicated topology akin to that of the Mandelbrot set studied in complex dynamics. In joint work with J. Holt, Bromberg has shown that this complicated behavior persists even if the deformation space is restricted to metrics on a 3-manifold of fixed homeomorphism type generalizing work of McMullen. Complex projective structures naturally appear in both of these topics. Bromberg also plans to study projective structures with a given discrete holonomy representation. This project lies in the field of low dimensional topology and geometry. The main objects studied are three-manifolds which are spaces with one more dimension than a surface (a 2-manifold). Three-manifolds, such as the space that we live in, are some of the most basic objects in mathematics yet there are many simple questions about them that we cannot answer. A metric is a way of measuring distance on a 3-manifold and a hyperbolic metric is one of a special class of metrics such that at every point and in every direction the geometry locally looks the same. The work of W. Thurston has shown that "most" 3-manifolds carry a hyperbolic metric and earlier work of Ahlfors, Bers and others has shown that an open 3-manifold that has an infinite volume hyperbolic metric will in fact have many hyperbolic metrics. There is a vast program to understand all such metrics of which this project is a piece.