Low-Dimensional Topology and Subdivision Rules

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Proposal: DMS-9971783PI: William FloydAbstract: This project is an attempt to prove the conjecture that a negatively curved group whose space at infinity is the 2-sphere has a properly discontinuous, cocompact, isometric action on real hyperbolic 3-space. This is a major piece of Thurston's Geometrization Conjecture, which states that the interior of every compact 3-manifold has a canonical decomposition into geometric pieces. Previous work of the investigator and his collaborators has reduced the conjecture to a difficult problem (proving conformality) about recursive tilings of the 2-sphere. The recursive structures of the tilings come from subdivision rules. From a subdivision rule one can define a branched surface and a self-map of this branched surface. A major aim is to prove that conformality of a tiling family is equivalent to the existence of a fixed point on the induced map of the Teichmueller space of the branched surface, and then to understand when this map has a fixed point. In the special case that the branched surface is a 2-sphere, this is essentially what is done in proving Thurston's classification theorem for critically finite branched maps of the 2-sphere. A second aim is to understand further this connection between conformal subdivision rules and rational maps. A third aim is to develop the theory of twisted face pairing 3-manifolds. This theory arose as a source of examples for testing the conjecture, but it has other potential applications in geometric group theory and 3-manifold topology.This project is concerned with recursive families of tilings of the plane and the 2-sphere. Each family is determined by an initial tiling and a subdivision rule which are encoded by a finite amount of combinatorial information. A key problem is to determine when the limiting tiles, which may have fractal boundaries, stay "almost round". In special cases these families of tilings correspond to rational maps (quotients of polynomials with complex coefficients) on the 2-sphere. Through this correspondence, work of Thurston on the topological classification of rational maps suggests a promising approach to answering this problem. A principal aim of the project is to follow this approach to completion. If successful, the project would be a major step in proving Thurston's Geometrization Conjecture, which states that 3-dimensional spaces (more precisely, compact 3-manifolds) can be naturally decomposed into geometric pieces. There are potential applications in complex variables, geometric group theory, 3-dimensional topology, and in the theory of biological cell growth.