Structure and Deformation in Low-Dimensional Topology

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Geometry and topology in two and three dimensions display a particular richness of structure where many different fields of mathematics come together, including classical complex analysis, combinatorial topology, geometry, dynamical systems, and the theory of groups. When three-dimensional spaces (manifolds) are glued together along their two-dimensional boundaries, the properties of the resulting space depend in complex ways on the properties of the gluing maps. The gluing maps themselves form a rich algebraic structure known as a mapping class group; this research project explores a number of settings where this structure comes into play. A particular role is played by a special geometric property of the mapping class group known as relative hyperbolicity, which has enabled the fine geometric structure of three-dimensional manifolds to be teased out and analyzed. Some of the previous advances in this area have yielded general statements but not concrete estimates, and in a number of areas the investigator expects to be able to sharpen current understanding and compute concrete bounds. The focus, nevertheless, is to deepen conceptual understanding of the way that two-dimensional and three-dimensional manifolds fit together. The project investigates aspects of the interaction between topology of surfaces and three-manifolds, with connections to deformation spaces of geometric structures and geodesic flows in Teichmuller space. A common thread in the work is the hierarchical structure of mapping class groups and the phenomenon of relative hyperbolicity via curve complexes. In a project on fibered hyperbolic three-manifolds, the investigator will explore the "subsurface profiles" of the set of all fibrations, namely the pattern of projections to fiber subsurfaces of the stable and unstable foliations associated to the monodromy maps. The goal here is to obtain uniform (genus independent) descriptions and to relate them to the combinatorial structure of veering triangulations. In Teichmuller theory, the investigator will pursue a conjectural description of the Weil-Petersson geodesic flow on the moduli space of a surface, developing new tools with which to analyze the combinatorial patterns or itineraries of geodesic rays. The investigator will study the skinning map defined by Thurston on the Teichmuller space of the boundary of a hyperbolic three-manifold. He will work toward a better quantitative control of the diameter bounds known for such maps in the acylindrical case. In the general case he will extend previous work that established relative bounded image results, to complete the proof of a claim of Thurston on iterates of the skinning-and-gluing maps in the setting where a gluing yields an atoroidal manifold. This work will also connect to a project on establishing new uniform bi-Lipschitz models for hyperbolic three-manifolds.