Geometry, algebra, and analysis of moduli of hyperbolic manifolds

该项目探索了双曲流形的模空间的几何和拓扑，它们的基本群的代数，以及在邻近区域的应用。中心焦点是一个封闭曲面的映射类群的几何和代数，特别是子群的几何和代数行为和这个群的无限补全，以及它们所作用的空间。该项目的大部分内容都是通过双曲几何的镜头来观察的，其定理和技术在映射类组上投下了诱人的轮廓。项目的核心分为三个部分：(1)与C. Leininger正在进行的与Gromov粗夸张化问题相关的项目的延续，以理解映射类群的凸紧子群；(2)推广了PI、J. Brock和C. Leininger在具有双曲基群的曲面上产生映射类群和面束的纯伪anosov曲面子群的方案；(3)在PI先前工作的基础上，致力于完成M. Boggi的程序，以建立映射类曲面群的同余子群性质。模空间是一个几何对象的集合，它本身也是一个几何对象。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的排列。人们可以使用单个塔之间的距离来定义两个塔之间的距离，而塔的排列集合本身就成为一个几何对象，一个排列的“空间”。地理限制限制了塔的可行配置，了解可行配置空间的几何结构可以直接影响哪种配置提供最佳的网络覆盖。本课题研究双曲流形的模空间，其中地球上手机发射塔的构型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，创造了一种信息反馈回路，相互通知双曲流形及其模空间的研究。这个类比正是这个项目的核心。

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.