Hyperbolic Manifolds and Their Groups

三流形是一个空间，在这个空间中，物体可以在三个不同的垂直方向上移动。我们居住的宇宙是一个三维流形，其全球结构我们还不清楚。由于瑟斯顿、佩雷尔曼和莫斯托的强大定理，我们确实知道流形的几何(角度、距离和曲率的测量)与其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决定的定量理解。这个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，这个项目寻求在涉及负曲线三流形及其基本群的几何的几个基本问题上取得进展。一个问题涉及对Dehn手术下几何变化的定量控制，包括对整容手术猜想的应用。第二个问题涉及到理解克莱因群，在克莱因群中，两个独立的元素将一个点移动一小段距离，并应用于界定马古利斯常数。第三个问题涉及对三个流形小组如何作用于CAT(0)立方体复合体的定量理解，着眼于为Cannon猜想开发工具。这一奖项反映了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。

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.