Orbit Closures in Moduli Spaces of Surfaces and Surface Subgroups of Mapping Class Groups

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

Moduli spaces of surfaces play a central role throughout mathematics and theoretical physics. Each such moduli space can be thought of as a universe of all possible shapes that a surface can take. Just as the laws of gravity prescribe the orbits of the planets and stars of our universe, a different set of laws, called the GL+(2,R) action, prescribe orbits of surfaces in these universes of all possible surfaces. The first part of this project aims to obtain classification results for these orbits. Such classification results would provide a table of possible behaviors, and would give deep information about the surfaces themselves. This information could be used in more concrete mathematical and physical applications involving dynamics and polygonal shapes. The second part of this project aims to give insight into the mapping class group, which encodes all ways of wrapping a surface onto itself, as well as four dimensional shapes called surface bundles. Aspects of this research provide an ideal training ground for undergraduate, graduate, and postdoctoral students, and connect to the principal investigator's ongoing efforts to further popularize Maryam Mirzakhani's inspiring work in this area. Specifically, the PI proposes to engage in research at the intersection of Teichmueller theory and dynamics, in two related programs. The first program concerns GL+(2,R) orbit closures of quadratic and Abelian differentials, also known as (half) translation surfaces. The principal investigator proposes to classify orbit closures that have large rank, by employing recent theorems concerning cylinder deformations and degenerations developed by the principal investigator and his coauthors, as well as the seminal results of Eskin-Mirzakhani-Mohammadi. In the second program, the principal investigator proposes to investigate convex cocompact surface subgroups of the mapping class group, employing dynamical and geometric results concerning the Teichmueller geodesic flow. The existence of such surface subgroups is known to be equivalent to the existence of surface bundles over surface with Gromov hyperbolic fundamental group. 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.