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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上移动的空间。我们所居住的宇宙是一个三维流形，其整体结构我们还不了解。由于瑟斯顿、佩雷尔曼和莫斯托的有力定理，我们知道流形的几何（角度、距离和曲率的测量）与它的大尺度结构密切相关。在这一点上缺少的是对几何和大规模拓扑如何相互确定的定量理解。本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学的几个基本问题上取得进展。一个问题涉及定量控制的几何形状的变化下德恩手术，包括应用到整容手术猜想。第二个问题涉及到理解克莱因群，其中两个独立的元素移动一个点的一个小距离，与应用程序的边界马古利斯常数。第三个问题涉及对三元组作用于CAT（0）立方体复合体的方式的定量理解，着眼于开发Cannon猜想的工具。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。