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

几何学主要研究流形(多维形状)及其内在性质，如流形上两点之间的体积、曲率和曲线长度。在这一领域中，了解给定流形的对称性对于研究它的其他几何性质起着关键作用。这些对称被编码在称为基本群的代数结构中；这个项目的目的是研究这个群与几何之间的联系。具体地说，在双曲流形中，有一类特殊的“算术”，它往往是最对称的，其基本群与数论有很强的联系。这个项目的目的是利用几何和动力学中的新技术来研究双曲流形的基本群，试图理解这样的群何时是算术的，以及算术性(或缺乏算术性)对相关流形的几何的影响。这个项目的更广泛的影响包括与本科生的合作。更具体地说，这个研究项目的总体目标有两个--更好地理解双曲流形的分类及其测地几何，并为探索有限体积实流形、复流形、四元数流形和Cayley双曲流形的基本群的刚性现象建立一个健壮的框架。主要研究者最近取得了一系列的进展，促进了几何、群论和动力学技术的发展，以了解通过粘合算术流形的子流形建立的流形的测地几何，以及实双曲空间的等距群中的格的超刚性风格技术的发展。该项目计划继续开发这些着眼于几何应用的新技术。具体地说，该项目将解决以下广泛的主题：1)了解低维和高维双曲流形及其测地线子流形的构造，2)进一步开发秩1格超刚性结果的一般框架，以及3)尝试利用秩1刚性的最新进展作为一种机制来理解复杂双曲格子的完整性以及四元数和Cayley双曲空间的算术性。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.