Subdivision Rules and 3-Manifold Topology

Virginia Polytechnic Institute And State University, Blacksburg VA

Investigators

Abstract

DMS-0203902 William J. Floyd This project is an attempt to resolve the hyperbolic case of Thurston's Geometrization Conjecture. Specifically, the goal is to resolve the conjecture that a Gromov-hyperbolic group with space at infinity a 2-sphere has a cocompact, properly discontinuous action on hyperbolic 3-space. The investigator and his collaborators are approaching the conjecture from the point of view of proving conformality of certain recursive sequences of tilings on the space at infinity of a Gromov-hyperbolic group. Previous work has indicated that conformality of a recursive sequence of tilings might follow from finding an invariant conformal structure for a branched surface associated to the recursive structure. This possibility arose from a connection between the recursive structures and rational maps, and Thurston's classification theorem for critically finite branched maps of the 2-sphere gives insight into how the theory might develop. Multiple approaches are planned for finding an invariant conformal structure. Further work is also planned on twisted face-pairing 3-manifolds. Much of the basic theory of twisted face pairings has been completed, but some questions remain which are central to further developments of the theory. Significant progress here could help the main part of the project, since twisted face pairings are a good source of test examples for the conjecture stated above. The immediate focus of this proposal is on sequences of planar tilings. Given an initial tiling and a combinatorial rule for subdivision, one recursively obtains a sequence of subdivisions of the initial tiling. The goal is to understand when these combinatorial subdivisions can be realized geometrically so that the tiles stay "almost round" at all stages of the sequence. This problem is interesting in its own right, but it is being studied here as part of a deeper problem. It is a key feature of a program of the investigator and his collaborators to resolve the hyperbolic case of William P. Thurston's Geometrization Conjecture. The Geometrization Conjecture, which is the central outstanding problem in low-dimensional topology (it includes the Poincare conjecture as a special case), states that every compact 3-manifold can be naturally subdivided into geometric pieces. The techniques being developed for approaching this have potential applications in other disciplines.