Mathematical Sciences: Geometric Topology and the Fundamental Group

数学科学：几何拓扑和基本群

• 批准号: 9506725
• 负责人: James Cannon
• 金额: $ 9.32万
• 依托单位: Brigham Young University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-07-15 至 1998-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9506725&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Geometric; Topology; Fundamental

9506725 Cannon This project asks, "Does variable negative curvature imply constant negative curvature in dimension three?" More precisely, the project seeks a proof that a negatively curved (Gromov hyperbolic) group with 2-sphere at infinity acts coaompactly, isometrically, and properly discontinuously on hyperbolic 3-space. This goal would constitute a major step forward in establishing Thurston's important geometrization conjecture for 3-manifolds and would also have important consequences in the study of Kleinian groups, negatively curved groups, conformal mapping, and Riemannian geometry. Prior work of the principal investigator and coworkers has reduced the problem to the study of discrete conformal mapping problems about finite subdivision rules in the plane. Secondary, but related, problems concern algorithmic techniques with negatively curved or almost convex groups. All of geometry and its applications takes place within mathematical models or "spaces." Topology seeks to classify these models and to understand their chief local and global properties. The most important of these models are the "manifolds," the locally Euclidean spaces. Mathematicians have to great effect long since classified the 2-dimensional manifolds and have a corresponding valuable, yet conjectural, picture of 3-dimensional manifolds, called "Thurston's geometrization conjecture." The project seeks a proof of one piece of that conjecture, namely that 3-manifolds that in the large behave like negatively curved spaces can in fact be smoothed so as to have constant negative curvature in the small. An affirmative answer would reduce many problems about 3-manifolds to well-developed techniques involving matrix theory, algebra, and analysis, with corresponding economies in mathematical physics and elsewhere that these models are used. ***

9506725加农炮这个项目提出的问题是：“可变的负曲率是否意味着三维空间中的恒定负曲率？”更确切地说，该项目试图证明一个负弯曲的(格罗莫夫双曲群)群在无穷远的2球面上作用于双曲3-空间上的余紧、等距且适当的间断作用。这一目标将是在建立瑟斯顿关于3-流形的重要几何化猜想方面向前迈出的重要一步，也将在Klein群、负曲群、保形映射和黎曼几何的研究中产生重要的结果。主要研究者和同事的前期工作已将问题归结为关于平面上有限细分规则的离散保角映射问题的研究。次要但相关的问题涉及负曲或几乎凸群的算法技术。所有的几何学及其应用都发生在数学模型或“空间”中。拓扑学试图对这些模型进行分类，并了解它们的主要局部和全局属性。这些模型中最重要的是“流形”，即局部欧几里德空间。数学家们很早以前就对二维流形进行了分类，并对三维流形有了一幅有价值但却是猜想的图景，称为“瑟斯顿几何化猜想”。该项目试图证明这一猜想中的一项，即在大的空间中表现为负曲线空间的3-流形实际上可以被平滑，从而在小的空间中具有恒定的负曲率。一个肯定的答案将把许多关于三维流形的问题归结为涉及矩阵理论、代数和分析的成熟技术，以及在数学物理和其他使用这些模型的地方的相应经济。***