Hyperbolic Manifolds and Their Moduli Spaces

在数学中，人们经常面临着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，变形这个物体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它来理解几何形状。这方面的典型例子是黎曼曲线的模空间，这是一个空间，其每个点代表一个称为黎曼曲面的二维物体。 该空间捕获了可以使这些表面变形的所有方式。 拟议的研究将探索这个空间的几何，它的类似物在某些三维空间的变形称为双曲流形，以及它们的相互关系。它也包含适合研究生的子项目。该项目福尔斯分为两个部分：双曲三流形的变形理论;和曲面的映射类群的子群的几何和代数性质的研究，其相关的曲面丛，和profinite完成。第一个主要是关于理解一个称为蒙皮映射的函数，它是瑟斯顿在他关于三流形的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了解三维流形的几何及其与拓扑的关系。 特别是，PI将继续与合作者Bromberg和Minsky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整化体积概念的关系。 第二部分是关于几何的研究子群的映射类群体和他们的关系几何的四流形合作与莱宁格。 第二部分也将继续PI的工作在profinite方面的映射类groups，着眼于伊万诺夫的一致性subgroup problem.This award reflects NSF's法定使命，并已被认为是值得的支持，通过评估使用基金会的智力价值和更广泛的影响审查标准.

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.