Geodesic Submanifolds, Rigidity, and Other New Phenomena in Rank 1

Temple University, Philadelphia PA

Investigators

Abstract

Lattices in Lie groups are a sweeping generalization of the integers inside the real numbers. The deep connections between the integers and a broad range of mathematical areas from geometry to number theory have direct analogues for lattices in Lie groups, and the way in which lattices in Lie groups sit at the interface of so many fields has made them of fundamental importance since the late 19th century. The primary motivation for this project is to deepen our understanding of these connections, in particular for hyperbolic manifolds and their close relatives. Hyperbolic geometry is the "negatively curved" counterpart to classical Euclidean geometry (zero curvature) and spherical geometry (positive curvature), and hyperbolic manifolds are poorly understood compared with their Euclidean and spherical counterparts. Relating hyperbolic manifolds to discrete subgroups of Lie groups provides access to a wide spectrum of mathematical tools, including algebra and dynamics, that have borne significant recent fruit, including recent work of the PI with U. Bader, D. Fisher, and N. Miller that gives a completely geometric characterization of which hyperbolic manifolds that are "arithmetic" (a family of immense significance, but one for which the definition is purely algebraic and not obviously tied to the geometry of the manifold). The overarching goal of this project is to use techniques from dynamics, geometry, algebraic geometry, and number theory to further our understanding of hyperbolic manifolds and closely-related generalizations called "locally symmetric spaces". The project provides support for graduate students allowing them time for research, collaboration, and travel to conferences. It will also support the PI's continued development of Inquiry-Based Learning tools for undergraduate teaching and organizing events like summer schools and conferences focused on graduate student professional development This project aims to understand the geometry and topology of locally symmetric spaces, particularly real and complex hyperbolic manifolds, inspired by the fundamental problems in low-dimensional topology and geometric group theory that have dominated the fields since the pivotal work of Thurston and Gromov. On the one hand, it is of significant interest to learn the extent to which the Gromov-Thurston program pushes into this more general setting. More importantly, the questions studied in dimensions 2 and 3 for the last forty years are closely related to classical problems about discrete subgroups of Lie groups, and real and complex hyperbolic lattices are precisely the cases where many of the basic questions remain open, e.g., (non)triviality of Betti numbers and problems related to (non)arithmeticity. Particular problems to be studied include analogues of the Margulis superrigidity theorem in rank one, building on the PI's recent breakthroughs with Bader-Fisher-Miller, and understanding complex hyperbolic lattices using techniques from algebraic geometry and continuing/generalizing the PI's work with D. Toledo on connections between residual finiteness of central extensions of complex hyperbolic lattices and constructions of interesting new smooth projective algebraic varieties admitting metrics of negative curvature. 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.