Geometry, Topology, and Rank One Lattices

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Lattices in Lie groups are a wide-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. There is a special class of lattices that are called "arithmetic", where the connections to number theory are particularly strong, and this an overarching goal of this project is to use techniques from dynamics and geometry to understand when a lattice is arithmetic, and moreover deepen our understanding of the geometric consequences of arithmeticity. The project provides summer support for graduate students allowing them time for research and collaboration.More specifically, 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., Betti numbers and problems related to (non)arithmeticity. Particular problems to be studied include geometric characterizations of arithmeticity, analogues of the Margulis superrigidity theorem in rank one, and understanding complex hyperbolic lattices using techniques from algebraic geometry.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.