Hyperbolic geometry and mapping class groups

曲面和三维流形是重要的研究对象，不仅因为它们包含丰富的数学，而且因为它们是出现在真实的世界中的基本对象。研究曲面和三维流形的一种方法是给它们几何。当这样做的时候，自然会导致双曲几何，因为只要一个看过去的最简单的例子几何的“大多数”曲面和3-流形是双曲的。双曲几何是三大经典几何之一。另外两个是我们更熟悉的欧几里德几何，或平面几何，一张桌子或一张纸的几何和网球或地球的球面几何。双曲几何，虽然不太熟悉，局部是模仿喇叭或长号的喇叭端，它是双曲几何的研究中心，在这个建议。这个项目研究几个相关的主题。投影复合体是PI与M. Beststanbul和K.藤原它最初用于研究映射类组，但已证明也适用于其他组。该项目的一个方面是继续研究该对象及其应用。该项目的另一个主要焦点是双曲三维流形的研究。特别是PI将继续与J. Brock，R. Canary和C. Lecuire，双曲三维流形的变形空间拓扑的研究。

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.