Problems in geometry, topology, and group theory

To understand complicated objects, it is useful to know how they are built from simpler pieces. For example, the individual parts of a car are relatively simple, but assemble into a complex and powerful machine; sectional images from MRI can be used to reconstruct a picture of the human body that carries enough information to diagnose medical problems. Studying objects as the amalgam of simpler pieces has great utility in mathematics. This project concerns complicated spaces that can be built out of simpler ones, namely surfaces, like the surface of a ball or a doughnut (as well as more complicated surfaces). The instructions for assembling the pieces are described by a mathematical object called the mapping class group, which carries all the information necessary to build certain spaces from surfaces. The PI will work with a team of PhD students, postdocs, and colleagues and investigate the kinds of spaces that can be built from surfaces, and their geometric, algebraic, and analytic features. This project involves the study of surfaces of finite and infinite type, their mapping class groups, and geometric features of manifolds and bundles we can understand from these. The PI, together with his students, postdocs, and collaborators, will focus on the following themes: (1) Convex cocompact and geometrically finite subgroups of the mapping class group for finite type surfaces and the geometry of the associated extension groups/surface bundles. (2) The hyperbolic geometry of depth-one foliations via mapping tori of end-periodic homeomorphism. (3) The geometric group theory of ``medium size” mapping class groups naturally containing end-periodic homeomorphisms. (4) Relations amongst pseudo-Anosov monodromies for fibered classes in a fixed hyperbolic 3-manifold. (5) Decoding geometry of billiard tables from the symbolic coding of its billiard flow. The PI will continue his investigations of these themes, and probe the intricate ways they interact with each other. 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.