Geometry of Mapping Class Groups and Surface Bundles

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Surfaces are two dimensional spaces which play a fundamental role in many areas of mathematics, especially in topology, geometry, and dynamics. Surfaces can be flat, like a piece of paper, or curved, like the outside of a ball, a donut, or a saddle, and the various shapes they can take often strongly constrain the shapes of the higher dimensional manifolds in which they inevitably occur. An outstanding and ubiquitous example of this phenomenon is a surface bundle, which, like a donut, is a higher dimensional manifold which can be sliced so that the cross-sections are surfaces. Unlike a donut, however, as one moves through most surface bundles, the surface cross-sections can twist and deform in complicated ways. This twisting is encoded in the geometry of the mapping class group, which, among other things, is the collection of all symmetries of the space of shapes that a surface can take.The central aim of this research program is to develop new tools for deepening our understanding of the geometry of the mapping class group and its connection to surface bundles, including by analyzing the geometry of important classes of 4-dimensional bundles coming from dynamics. In more detail, this project will investigate the coarse geometry of the mapping class group, Teichmuller space, and surface bundles over surfaces using the tools of geometric group theory. A unifying aspect of the various parts of the project is the hierarchical nature of these objects and the hyperbolicity of the curve complex. The PI will employ ideas from CAT(0) cubical geometry to study the local coarse geometry of the mapping class group. Using related ideas, the PI will investigate whether the mapping class group satisfies certain conjectures from K-theory. In another direction, the PI will study surface bundles over Teichmuller curves with the aim of developing a new theory of geometrical finiteness for surface bundles. Finally, the PI will study various algorithmic problems in the mapping class group, some of which have applications to surface bundles and symplectic topology.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.