Low-Dimensional Topology and Subdivision Rules

提议：DMS-9971783PI：William Floyd摘要：这个项目试图证明一个猜想，即一个在无穷远的空间为2-球面的负曲群在实双曲3-空间上有适当的间断、余紧、等距作用。这是瑟斯顿的几何化猜想的一个主要部分，该猜想指出每个紧凑的3-流形的内部都有一个到几何片的正则分解。研究人员和他的合作者以前的工作已经将猜想简化为关于2-球面的递归平铺的困难问题(证明一致性)。平铺的递归结构来自细分规则。根据细分规则，可以定义分支曲面和该分支曲面的自映射。主要目的是证明瓦片族的一致性等价于分支曲面的Teichmueller空间的诱导映射上存在不动点，从而理解该映射何时存在不动点。在分枝曲面是2-球面的特殊情况下，这是证明2-球面的临界有限分枝映射的瑟斯顿分类定理的实质。第二个目的是进一步了解保角细分规则和有理映射之间的这种联系。第三个目标是发展扭面配对3-流形理论。这个理论是作为检验猜想的例子而出现的，但它在几何群论和3-流形拓扑学中也有其他潜在的应用。每个家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。一个关键问题是确定限制瓷砖的时间，这些瓷砖可能有分形的边界，保持“几乎是圆形的”。在特殊情况下，这些平铺族对应于2-球面上的有理映射(具有复系数的多项式的商)。通过这种对应，瑟斯顿在有理映射的拓扑分类方面的工作提出了一种很有希望的方法来回答这个问题。该项目的一个主要目标是遵循这一方法完成。如果成功，这个项目将是证明瑟斯顿的几何化猜想的重要一步，该猜想声称三维空间(更准确地说，是紧致的3-流形)可以自然地分解成几何碎片。在复变量、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.