Deformations of hyperbolic 3-manifolds

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Proposal: DMS-0072062Abstract:Professor Bromberg plans to study spaces of hyperbolic metrics on a fixed3-manifold. Two types of questions will be investigated. The first ofthese is a study of hyperbolic cone-manifolds. Finite volumecone-manifolds arise naturally as a link between cusped hyperbolic3-manifolds and closed hyperbolic 3-manifolds. Bromberg's goal is toparameterize spaces of infinite volume hyperbolic cone-manifolds and usethis parameterization to understand non-singular hyperbolic structures.The second question involves the topology of deformation spaces ofcomplete, infinite volume metrics on open 3-manifolds. Anderson and Canaryhave shown that the space of such metrics on a manifold of fixed homotopytype has a very complicated topology akin to that of the Mandelbrot setstudied in complex dynamics. In joint work with J. Holt, Bromberg hasshown that this complicated behavior persists even if the deformationspace is restricted to metrics on a 3-manifold of fixed homeomorphism typegeneralizing work of McMullen. Complex projective structures naturallyappear in both of these topics. Bromberg also plans to study projectivestructures 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 onemore dimension than a surface (a 2-manifold). Three-manifolds,such as the space that we live in, are some of the most basicobjects in mathematics yet there are many simple questions about them thatwe cannot answer. A metric is a way of measuring distance on a 3-manifoldand a hyperbolic metric is one of a special class of metrics such that atevery point and in every direction the geometry locally looks the same.The work of W. Thurston has shown that "most" 3-manifolds carry ahyperbolic metric and earlier work of Ahlfors, Bers and others has shownthat an open 3-manifold that has an infinite volume hyperbolic metric willin fact have many hyperbolic metrics. There is a vast program tounderstand all such metrics of which this project is a piece.