Hyperbolic geometry and mapping class groups

曲面和3流形是重要的研究对象，不仅因为它们包含丰富的数学内容，而且因为它们是现实世界中出现的基本对象。研究曲面和3流形的一种方法是赋予它们几何学。当这样做的时候，一个人很自然地导致双曲几何，因为只要一个人看了最简单的例子，“大多数”曲面和3流形的几何是双曲的。双曲几何是三大经典几何之一。另外两种是我们更熟悉的欧几里得几何，即桌面或纸张的平面几何，以及网球或地球的球形几何。双曲几何，虽然不太熟悉，但局部是模仿小号或长号的喇叭口，这是双曲几何，是本提案研究的中心。这个项目研究了几个相关的主题。投影综合体是由PI与M. Bestvina和K. Fujiwara合作开发的。它最初用于研究映射类组，但也显示出对其他组的应用。该项目的一个方面是继续对该对象及其应用进行研究。该项目的另一个主要重点是双曲3流形的研究。特别是PI将继续与J. Brock， R. Canary和C. Lecuire合作，研究双曲型3流形的变形空间拓扑。

Surfaces and 3-manifolds are important objects of study not only because of the rich mathematics they contain but also because they are fundamental objects that appear in the real world. One approach for studying surfaces and 3-manifolds is to give them geometry. When doing so one is naturally lead to hyperbolic geometry because as soon as one looks past the simplest examples the geometry of "most" surfaces and 3-manifolds is hyperbolic. Hyperbolic geometry is one of the three classical geometries. The other two are the more familiar Euclidean, or flat, geometry of a table top or piece of paper and the spherical geometry of a tennis ball or the earth. Hyperbolic geometry, while less familiar, locally is modeled on the flaring end of a trumpet or trombone and it is hyperbolic geometry that is at the center of the research in this proposal.This project studies several related topics. The projection complex is a construction developed by the PI in collaboration with M. Bestvina and K. Fujiwara. It was originally used to study the mapping class group but has shown to have applications to other groups as well. One aspect of this project is to continue the study of this object and its applications. The other main focus of the project is the study of hyperbolic 3-manifolds. In particular the PI will continue, in collaboration with J. Brock, R. Canary and C. Lecuire, the study of the topology of deformation spaces of hyperbolic 3-manifolds.