Hyperbolic Manifolds and Their Groups

双曲流形及其群

基本信息

批准号：
1907708
负责人：
David Futer
金额：
$ 24.34万
依托单位：
Temple University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-07-01 至 2023-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1907708&HistoricalAwards=false
关键词：
Hyperbolic Manifolds Their Groups

项目摘要

A three-manifold is a space where an object can move around in three distinct perpendicular directions. The universe that we inhabit is a three-manifold whose global structure we do not yet understand. Thanks to powerful theorems by Thurston, Perelman, and Mostow, we do know that the geometry of a manifold (measurements of angles, distances, and curvature) is closely tied to its large-scale structure. What is missing at this point is a quantitative understanding of how geometry and large-scale topology determine one another. This project seeks quantitative information of this nature. It contains suitable sub-projects for graduate students. More specifically, this project seeks to make progress on several fundamental questions involving the geometry of negatively curved three-manifolds and their fundamental groups. One question involves quantitative control on the change in geometry under Dehn surgery, including applications to the cosmetic surgery conjecture. A second question involves understanding Kleinian groups in which two independent elements move a point by a small distance, with applications to bounding the Margulis constant. A third question involves a quantitative understanding of the way in which three-manifold groups act on CAT(0) cube complexes, with an eye toward developing tools for the Cannon conjecture.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.

三流形是一个物体可以在三个不同的垂直方向移动的空间。我们居住的宇宙是一个三流形，我们还不了解其整体结构。得益于瑟斯顿、佩雷尔曼和莫斯托强大的定理，我们确实知道流形的几何形状（角度、距离和曲率的测量）与其大规模结构密切相关。目前缺少的是对几何和大规模拓扑如何相互决定的定量理解。该项目寻求这种性质的定量信息。它包含适合研究生的子项目。更具体地说，该项目旨在在涉及负弯曲三流形及其基本群的几何形状的几个基本问题上取得进展。其中一个问题涉及对德恩手术下几何形状变化的定量控制，包括在整容手术猜想中的应用。第二个问题涉及理解克莱因群，其中两个独立元素将一个点移动一小段距离，并应用于限制马古利斯常数。第三个问题涉及对三流形群作用于 CAT(0) 立方体复合体的方式的定量理解，着眼于为 Cannon 猜想开发工具。该奖项反映了 NSF 的法定使命，并被认为值得通过以下方式获得支持：使用基金会的智力价值和更广泛的影响审查标准进行评估。