Hyperbolic Manifolds, Geodesic Submanifolds, and Rigidity for Rank-1 Lattices

Geometry is broadly focused on studying manifolds (multi-dimensional shapes) and their intrinsic properties, such as volume, curvature, and lengths of curves between two points on the manifold. In this field, understanding symmetries of a given manifold plays a key role in studying its other geometric properties. These symmetries are encoded in an algebraic construction called the fundamental group; this project aims at studying the connections between this group and geometry. Specifically, among hyperbolic manifolds there is a special class called "arithmetic" that tend to be the most symmetric and whose fundamental group has strong connections to number theory. This project aims to use new techniques in geometry and dynamics to study the fundamental group of hyperbolic manifolds in an attempt to understand when such a group is arithmetic and the ramifications of arithmeticity (or lack thereof) on the geometry of the associated manifold. Broader impacts of this project include work with undergraduates. More specifically, the overarching goal of this research project is twofold -- to better understand the classification of hyperbolic manifolds and their geodesic geometry and to build a robust framework for exploring rigidity phenomenon for fundamental groups of finite-volume real, complex, quaternionic, and Cayley hyperbolic manifolds. The principal investigator has recently made a series of advances that facilitate the development of geometric, group theoretic, and dynamical techniques for understanding the geodesic geometry of manifolds built by gluing submanifolds of arithmetic manifolds, as well as the development of superrigidity style techniques for lattices in the isometry group of real hyperbolic space. This project plans to continue to develop these new techniques with an eye toward geometric applications. Specifically, the project will address the following broad themes: 1) understanding constructions of both low- and high-dimensional hyperbolic manifolds and their geodesic submanifolds, 2) further developing a general framework for superrigidity results for rank-1 lattices, and 3) attempting to use recent advances in rank-1 rigidity as a mechanism to understand integrality of complex hyperbolic lattices and arithmeticity of quaternionic and Cayley hyperbolic spaces. 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.