Mathematical Sciences: Geometric Topology and the Fundamental Group

这个项目问：“可变负曲率是否意味着三维空间中的恒定负曲率？”更确切地说，该项目寻求证明负弯曲（Gromov双曲）群在无穷远处具有2球，在双曲三维空间上是紧的、等距的和适当不连续的。这一目标在建立瑟斯顿关于3流形的重要几何化猜想方面迈出了重要的一步，并对Kleinian群、负弯曲群、共形映射和黎曼几何的研究产生了重要的影响。在之前的工作中，主要研究者和同事将该问题简化为研究平面上有限细分规则的离散保形映射问题。次要但相关的问题涉及负弯曲或几乎凸群的算法技术。所有几何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”，局部欧几里得空间。数学家很早就对二维流形进行了分类，并提出了相应的有价值的、推测性的三维流形图，称为“瑟斯顿几何化猜想”。该项目寻求证明该猜想的一部分，即在大空间中表现为负弯曲空间的3-流形实际上可以被平滑，从而在小空间中具有恒定的负曲率。一个肯定的答案会将许多关于3流形的问题简化为涉及矩阵理论、代数和分析的成熟技术，以及使用这些模型的数学物理和其他地方的相应经济。＊＊＊

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. ***