Geometry, algebra, and analysis of moduli of hyperbolic manifolds

该项目探讨了双曲流形的模空间的几何和拓扑，其基本群的代数，以及在相邻领域的应用。中心重点是几何和代数的映射类组的一个封闭的表面，特别是几何和代数行为的子群和profinite完成这个群体，和空间上，他们的行动。这个项目的大部分内容都是通过双曲几何的透镜来观察的，双曲几何的定理和技巧为映射类组投下了诱人的轮廓。该项目的核心福尔斯分为三个部分：（1）与C. Leininger，与Gromov的粗双曲化问题有关，以理解曲面的映射类群的凸余紧子群;（2）继续PI，J. Brock和C。Leininger的工作，以产生映射类群的纯伪Anosov曲面子群和具有双曲基本群的曲面上的曲面丛; Boggi的程序建立了曲面映射类组的同余子群属性，建立在PI的先前工作之上。A模空间是一个几何对象的集合，它本身就是一个几何对象。一个实际的例子是收集地球表面上所有可能的手机信号塔排列。 人们可以使用各个塔之间的距离来限定两个塔布置之间的距离，并且塔布置的集合本身变成几何对象，布置的“空间”。地理约束限制了塔的可行配置，并且理解可行配置的空间的几何形状可以直接影响哪些配置提供最佳网络覆盖。该项目涉及双曲流形的模空间，其中地球上的手机塔的配置空间是一个特殊的例子。 在双曲流形的模空间和双曲流形本身之间有一个有趣的类比。 换句话说，在某种意义上，双曲流形的集合可以被粗略地认为是双曲流形本身，从而创建一种信息反馈回路，为双曲流形及其模空间的研究提供信息。这个类比是这个项目的核心。

The project explores the geometry and topology of moduli spaces of hyperbolic manifolds, the algebra of their fundamental groups, and applications to adjacent areas. The central focus is the geometry and algebra of the mapping class group of a closed surface, specifically the geometric and algebraic behavior of subgroups and profinite completions of this group, and spaces upon which they act. Much of the project is viewed through the lens of hyperbolic geometry, whose theorems and techniques cast a seductive profile upon the mapping class group. The core of the project falls into three parts: (1) continuation of an ongoing project with C. Leininger, relevant to Gromov's Coarse Hyperbolization Problem, to understand convex cocompact subgroups of mapping class groups of surfaces; (2) continuation of a program of the PI, J. Brock, and C. Leininger to produce purely pseudo-Anosov surface subgroups of mapping class groups and surface bundles over surfaces with hyperbolic fundamental group; (3) work devoted to completion of M. Boggi's program to establish the congruence subgroup property for mapping class groups of surfaces, building upon prior work of the PI.A moduli space is a collection of geometric objects that is itself a geometric object. A practical example is the collection of all possible arrangements of cell phone towers on the surface of the Earth. One may use the distances between individual towers to define a distance between two arrangements of towers, and the collection of arrangements of towers becomes a geometric object itself, a "space" of arrangements. Geographical constraints limit the feasible configurations of towers, and understanding the geometry of the space of feasible configurations can have direct bearing on which configurations provide the best network coverage. The project is concerned with moduli spaces of hyperbolic manifolds, of which the space of configurations of cell phone towers on the Earth is a special example. There is an intriguing analogy between moduli spaces of hyperbolic manifolds and the hyperbolic manifolds themselves. In other words, there is a sense in which a collection of hyperbolic manifolds may be roughly considered a hyperbolic manifold itself, creating a sort of information feedback loop reciprocally informing the study of both the hyperbolic manifolds and their moduli spaces. It is this analogy that lies at the heart of the project.