Large-Scale Geometry of Hyperbolic Groups and 3-Manifolds

该提案由两个项目组成，它们的动机是三维流形的几何形状。第一个项目使用拟对称同胚作为一个框架来理解双曲群的渐近几何。讨论了几个均匀化和刚性问题。中心问题是Cannon关于边界同胚于二球的双曲群是双曲空间等距群中的格的猜想。这一猜想的解决将是向瑟斯顿的夸张化猜想迈出的重要一步。该提议的第二部分解决了弱夸张猜想，这是瑟斯顿猜想的另一个缺失的证明步骤。提出的攻击利用几何和动力学的表面层压在三个流形。数学中一个主要的开放问题是确定三维空间可能呈现的形状（拓扑）。在20世纪70年代，瑟斯顿提出了他著名的几何化猜想，这是对这样的空间应该如何分类的精确表述。该提案解决了瑟斯顿猜想的一部分，即所谓的夸张猜想。一般的攻击路线是将二维自相似分形球与每个三维空间联系起来，并将原始猜想简化为表明人们可以为这个相关的分形球找到“好的”坐标系。

DMS-0204506Bruce KleinerThe proposal consists of two projects which aremotivated by the geometry of three dimensional manifolds.The first project uses quasisymmetric homeomorphismsas a framework to understand the asymptotic geometryof hyperbolic groups. Several uniformization and rigidity problems are discussed. The central problemis Cannon's conjecture that a hyperbolic group withboundary homeomorphic to the two-sphere is a latticein the isometry group of hyperbolic space. A resolutionof this conjecture would be a major step toward Thurston'sHyperbolization conjecture. The second part of the proposal addresses the Weak hyperbolization conjecture,the other missing step in the proof of Thurston's conjecture.The proposed attack uses geometry and dynamics ofsurface laminations in three manifolds.One of the major open problems in Mathematics is to determine the possible shapes (topologies) that a three dimensional space might assume. In the 1970's, Thurston formulated his famousGeometrization conjecture, which is a precise statement abouthow the classification of such spaces should look. The proposal addresses a piece of Thurston's conjecture known as the Hyperbolization conjecture. The general line of attack is to associate a two dimensional self-similar fractal sphere with each three dimensional space, and reduce the originalconjecture to showing that one can find "good" coordinate systemsfor this associated fractal sphere.