Geometry, algebra, and analysis of moduli of hyperbolic manifolds

该项目探索双曲流形的模空间的几何和拓扑，它们的基本群的代数，以及在相邻区域的应用。中心焦点是闭曲面的映射类群的几何和代数，特别是子群和这个群的无限完备化的几何和代数行为，以及它们作用在的空间。这个项目的大部分是通过双曲几何的镜头来观察的，其定理和技术在映射类群上投下了诱人的轮廓。该项目的核心分为三个部分：(1)与C.Leininger正在进行的项目的继续，与Gromov的粗双曲化问题有关，以了解曲面映射类群的凸余紧子群；(2)PI，J.Brock和C.Leininger的程序的继续，以产生具有双曲基本群的曲面上的映射类群和曲面丛的纯伪Anosov曲面子群；(3)致力于完成M.Boggi的程序以建立映射曲面类群的同余子群性质的工作。模空间是几何对象的集合，它本身就是几何对象。一个实际的例子是收集地球表面所有可能的手机发射塔的排列。人们可以使用单个塔楼之间的距离来定义两个塔楼排列之间的距离，而塔楼排列的集合本身就变成了一个几何对象，一个排列的“空间”。地理限制限制了塔楼的可行配置，而了解可行配置空间的几何形状可以直接影响哪些配置提供最佳的网络覆盖。该项目涉及双曲流形的模空间，其中地球上的手机发射塔的配置空间是一个特例。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粗略地被认为是双曲流形本身，创建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。正是这种类比处于该项目的核心。

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.