Hyperbolic Manifolds and Their Moduli Spaces

在数学中，人们经常面临理解给定几何对象的问题。令人惊讶的是，为了更好地理解这个物体，改变它的几何形状往往是有利的。也就是说，一个人可以通过理解如何改变一个几何的东西来理解它。这方面的典型例子是黎曼曲线模空间，它的每个点代表一个二维物体，称为黎曼曲面。这个空间捕捉了所有可能使这些表面变形的方式。提出的研究将探索这个空间的几何形状，它在某些三维空间的变形空间中的类似物，称为双曲流形，以及它们的相互关系。它还包含适合研究生的子课题。本课题分为两部分：双曲型三流形的变形理论；并研究映射类曲面群及其相关曲面束和无限补全的子群的几何和代数性质。第一个主要是关于理解一个叫做蒙皮映射的函数，这个函数是瑟斯顿在他关于三流形几何化的工作中发现和研究的，它测量了双曲三流形变形的影响。更好地理解这个函数有助于阐明三流形的几何形状及其与拓扑的关系。特别是，PI将继续与合作者Bromberg和Minsky一起进行一个正在进行的项目，以建立该地图直径的有效界限及其与重归一化体积概念的关系。第二部分是与Leininger合作研究映射类群的子群几何及其与四流形几何的关系。第二部分还将继续PI在映射类群的无限方面的工作，并着眼于Ivanov的同余子群问题。该奖项反映了美国国家科学基金会的法定使命，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。

In mathematics one is often faced with the problem of understanding a given geometric object. Surprisingly often, it is advantageous to deform the geometry of this object in order to better understand it. That is, one may understand something geometric by understanding how one may change it. The prototypical example of this is Riemann's moduli space of curves, which is a space each of whose points represents a two-dimensional object called a Riemann surface. This space captures all of the ways one may deform these surfaces. The proposed research will explore the geometry of this space, its analogs in spaces of deformations of certain three-dimensional spaces called hyperbolic manifolds, and their interrelationships. It also contains suitable sub-projects for graduate students.The project falls into two parts: the deformation theory of hyperbolic three-manifolds; and the study of geometric and algebraic properties of subgroups of mapping class groups of surfaces, their associated surface bundles, and profinite completions. The first is principally concerned with understanding a certain function called the skinning map, which was discovered and studied by Thurston in his work on the geometrization of three-manifolds, and which measures the effects of deformations of hyperbolic three-manifolds. A better understanding of this function sheds light on the geometry of three-manifolds and its relation to topology. In particular, the PI will continue an ongoing project with collaborators Bromberg and Minsky to establish effective bounds on the diameter of this map and its relation to the notion of renormalized volume. The second part is concerned with the study of the geometry of subgroups of mapping class groups and their relation to the geometry of four-manifolds in collaboration with Leininger. The second part will also continue the PI's work in profinite aspects of mapping class groups, with a view toward Ivanov's congruence subgroup problem.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.