Geometry, algebra, and analysis of moduli of hyperbolic manifolds

几何、代数和双曲流形模分析

• 批准号: 1104871
• 负责人: AUTUMN KENT
• 金额: $ 15.37万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-01 至 2016-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1104871&HistoricalAwards=false
• 关键词: Geometry; algebra; analysis; moduli; hyperbolic

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.

该项目探索双曲流形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。中心焦点是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以及该群的有限完成，以及它们作用的空间。该项目的大部分内容都是通过双曲几何的透镜来看待的，双曲几何的定理和技术给映射类组带来了诱人的轮廓。该项目的核心分为三个部分：(1) 与 C. Leininger 正在进行的项目的延续，与格罗莫夫的粗双曲化问题相关，以理解曲面映射类组的凸协紧子组； (2) PI、J. Brock 和 C. Leininger 的程序的继续，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； (3) 致力于完成 M. Boggi 程序的工作，该程序建立了用于映射曲面类组的同余子组属性，建立在 PI 先前工作的基础上。模空间是几何对象的集合，其本身就是一个几何对象。一个实际的例子是地球表面手机信号塔所有可能排列的集合。 人们可以使用各个塔之间的距离来定义塔的两个布置之间的距离，并且塔的布置的集合本身变成几何对象，即布置的“空间”。地理限制限制了塔的可行配置，并且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网络覆盖。该项目涉及双曲流形的模空间，其中地球上手机信号塔的配置空间是一个特殊的例子。 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，从某种意义上说，双曲流形的集合可以粗略地视为双曲流形本身，从而创建一种信息反馈循环，相互告知双曲流形及其模空间的研究。这个类比是该项目的核心。