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The Deformation Space of a Hyperbolic 3-Manifold
Student-Run Geometry/Topology Seminar| Speaker: | Aaron Magid, University of Michigan |
| Location: | 2112 MSB |
| Start time: | Tue, Oct 16 2007, 1:10PM |
Given a hyperbolic 3-manifold M, we will define a topology on AH(M), the space of (marked) hyperbolic 3-manifolds homotopy equivalent to M. This space has connections with the Teichmuller space of a surface, quasiconformal maps on the sphere, and hyperbolic geometry. We will define all of these objects, discuss two equivalent definitions of AH(M), and characterize the interior of AH(M) as the set of convex cocompact hyperbolic 3-manifolds. Following the necessary background, we will outline a proof that (given some mild assumptions on M) the interior of AH(M) is homeomorphic to a product of Teichmuller spaces.
References
1. Richard Canary and Darryl McCullough. Homotopy Equivalences of 3-Manifolds and Deformation Theory of Kleinian Groups. Memoirs of the AMS, No. 812. Vol 172 (2004). Mostly Chapter 7.
2. R. Canary. "Introductory bumponomics: the topology of deformation spaces of hyperbolic 3-manifolds, 'fixed-point full automorphisms'." Available at http://www.math.lsa.umich.edu/canary
3. O. Lehto. Univalent Functions and Teichmuller Spaces, Springer-Verlag, 1987.
4. O. Lehto and K.I. Virtanen. Quasiconformal Mappings in the Plane, Springer-Verlag, New York, 1973.
5. W.P. Thurston. The Geometry and Topology of 3-Manifolds, lecture notes (mostly Chapter 8), 1980. Available at http://www.msri.org/communications/books/gt3m