Low-Dimensional Topology and Subdivision Rules

摘要：本课题试图证明无穷远空间为2球的负弯曲群在实双曲三维空间上具有适当的不连续、紧致、等距作用的猜想。这是瑟斯顿几何化猜想的一个主要部分，该猜想指出每个紧化3流形的内部都有正则分解成几何块。研究者和他的合作者之前的工作已经将这个猜想简化为一个关于2球的递归平铺的难题（证明一致性）。平铺的递归结构来自于细分规则。根据细分规则可以定义分支曲面和该分支曲面的自映射。主要目的是证明一个平铺族的一致性等价于分支曲面的Teichmueller空间的诱导映射上存在一个不动点，然后了解这个映射何时有不动点。在分支曲面为2球的特殊情况下，这实质上是在证明2球的临界有限分支映射的Thurston分类定理时所做的。第二个目标是进一步理解保形细分规则和理性映射之间的联系。第三个目标是发展扭曲面对3流形的理论。这个理论作为测试猜想的例子来源而出现，但它在几何群论和3流形拓扑中有其他潜在的应用。这个项目是关于平面和2球的递归族。每个族由初始平铺和细分规则确定，这些规则由有限数量的组合信息编码。一个关键的问题是确定限制瓷砖，它可能具有分形边界，保持“几乎是圆的”。在特殊情况下，这些族对应于2球上的有理映射（复数系数多项式的商）。通过这种通信，Thurston关于有理映射的拓扑分类的工作提出了回答这个问题的有希望的方法。该项目的主要目的是遵循这种方法完成。如果成功，该项目将是证明瑟斯顿几何猜想的重要一步，该猜想指出三维空间（更准确地说，紧凑的3-流形）可以自然地分解成几何块。在复杂变量、几何群论、三维拓扑学和生物细胞生长理论中有潜在的应用。

Proposal: DMS-9971783PI: William FloydAbstract: This project is an attempt to prove the conjecture that a negatively curved group whose space at infinity is the 2-sphere has a properly discontinuous, cocompact, isometric action on real hyperbolic 3-space. This is a major piece of Thurston's Geometrization Conjecture, which states that the interior of every compact 3-manifold has a canonical decomposition into geometric pieces. Previous work of the investigator and his collaborators has reduced the conjecture to a difficult problem (proving conformality) about recursive tilings of the 2-sphere. The recursive structures of the tilings come from subdivision rules. From a subdivision rule one can define a branched surface and a self-map of this branched surface. A major aim is to prove that conformality of a tiling family is equivalent to the existence of a fixed point on the induced map of the Teichmueller space of the branched surface, and then to understand when this map has a fixed point. In the special case that the branched surface is a 2-sphere, this is essentially what is done in proving Thurston's classification theorem for critically finite branched maps of the 2-sphere. A second aim is to understand further this connection between conformal subdivision rules and rational maps. A third aim is to develop the theory of twisted face pairing 3-manifolds. This theory arose as a source of examples for testing the conjecture, but it has other potential applications in geometric group theory and 3-manifold topology.This project is concerned with recursive families of tilings of the plane and the 2-sphere. Each family is determined by an initial tiling and a subdivision rule which are encoded by a finite amount of combinatorial information. A key problem is to determine when the limiting tiles, which may have fractal boundaries, stay "almost round". In special cases these families of tilings correspond to rational maps (quotients of polynomials with complex coefficients) on the 2-sphere. Through this correspondence, work of Thurston on the topological classification of rational maps suggests a promising approach to answering this problem. A principal aim of the project is to follow this approach to completion. If successful, the project would be a major step in proving Thurston's Geometrization Conjecture, which states that 3-dimensional spaces (more precisely, compact 3-manifolds) can be naturally decomposed into geometric pieces. There are potential applications in complex variables, geometric group theory, 3-dimensional topology, and in the theory of biological cell growth.