Subdivision Rules and 3-Manifold Topology

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DMS-0203902William J. FloydThis project is an attempt to resolve the hyperbolic case ofThurston's Geometrization Conjecture. Specifically, the goal is toresolve the conjecture that a Gromov-hyperbolic group with space atinfinity a 2-sphere has a cocompact, properly discontinuous action onhyperbolic 3-space. The investigator and his collaborators areapproaching the conjecture from the point of view of provingconformality of certain recursive sequences of tilings on the spaceat infinity of a Gromov-hyperbolic group. Previous work has indicatedthat conformality of a recursive sequence of tilings might followfrom finding an invariant conformal structure for a branched surfaceassociated to the recursive structure. This possibility arose from aconnection between the recursive structures and rational maps, andThurston's classification theorem for critically finite branched mapsof the 2-sphere gives insight into how the theory might develop.Multiple approaches are planned for finding an invariant conformalstructure. Further work is also planned on twisted face-pairing3-manifolds. Much of the basic theory of twisted face pairings hasbeen completed, but some questions remain which are central tofurther developments of the theory. Significant progress here couldhelp the main part of the project, since twisted face pairings are agood source of test examples for the conjecture stated above.The immediate focus of this proposal is on sequences of planartilings. Given an initial tiling and a combinatorial rule forsubdivision, one recursively obtains a sequence of subdivisions ofthe initial tiling. The goal is to understand when these combinatorialsubdivisions can be realized geometrically so that the tiles stay"almost round" at all stages of the sequence. This problem isinteresting in its own right, but it is being studied here as part ofa deeper problem. It is a key feature of a program of theinvestigator and his collaborators to resolve the hyperbolic case ofWilliam P. Thurston's Geometrization Conjecture. The GeometrizationConjecture, which is the central outstanding problem inlow-dimensional topology (it includes the Poincare conjecture as aspecial case), states that every compact 3-manifold can be naturallysubdivided into geometric pieces. The techniques being developed forapproaching this have potential applications in other disciplines.