Subdivision Rules and 3-Manifold Topology

DMS-0203902 William J. Floorman这个项目是试图解决Thurston的几何化猜想的双曲情况。具体地说，我们的目标是解决一个猜想，即一个空间在2-球面上的Gromov-双曲群在3-双曲空间上有一个余紧的，真不连续的作用。研究者和他的合作者们从证明Gromov-双曲群无穷远空间上的某些平铺递归序列的共形性的角度来处理这个猜想。以前的工作表明，一致性的递归序列的平铺可能遵循从找到一个不变的共形结构的分支surfacesociated递归结构。这种可能性来自递归结构和有理映射之间的联系，Thurston的2-sphere的临界有限分支映射的分类定理使我们深入了解了理论的发展。进一步的工作也计划在扭曲面pairing 3-流形。扭曲面配对的基本理论已经完成了很多，但仍存在一些问题，这些问题是扭曲面配对理论进一步发展的核心。这里的重大进展可能有助于项目的主要部分，因为扭曲面配对是上述猜想的测试示例的良好来源。给定一个初始镶嵌和一个组合细分规则，递归地得到初始镶嵌的细分序列。我们的目标是了解这些组合细分何时可以几何实现，以便瓷砖在序列的所有阶段都保持“几乎圆形”。这个问题本身就很有趣，但它是作为一个更深层次问题的一部分来研究的。这是一个程序的调查员和他的合作者解决双曲的情况下威廉P瑟斯顿的几何化猜想的一个关键特征。几何化猜想是低维拓扑学中的中心问题（它包括Poincare猜想作为特例），它指出每一个紧致3-流形都可以自然地细分为几何块。为接近这一点而开发的技术在其他学科中也有潜在的应用。

DMS-0203902William J. FloydThis project is an attempt to resolve the hyperbolic case ofThurston's Geometrization Conjecture. Specifically, the goal is toresolve the conjecture that a Gromov-hyperbolic group with space atinfinity a 2-sphere has a cocompact, properly discontinuous action onhyperbolic 3-space. The investigator and his collaborators areapproaching the conjecture from the point of view of provingconformality of certain recursive sequences of tilings on the spaceat infinity of a Gromov-hyperbolic group. Previous work has indicatedthat conformality of a recursive sequence of tilings might followfrom finding an invariant conformal structure for a branched surfaceassociated to the recursive structure. This possibility arose from aconnection between the recursive structures and rational maps, andThurston's classification theorem for critically finite branched mapsof the 2-sphere gives insight into how the theory might develop.Multiple approaches are planned for finding an invariant conformalstructure. Further work is also planned on twisted face-pairing3-manifolds. Much of the basic theory of twisted face pairings hasbeen completed, but some questions remain which are central tofurther developments of the theory. Significant progress here couldhelp the main part of the project, since twisted face pairings are agood source of test examples for the conjecture stated above.The immediate focus of this proposal is on sequences of planartilings. Given an initial tiling and a combinatorial rule forsubdivision, one recursively obtains a sequence of subdivisions ofthe initial tiling. The goal is to understand when these combinatorialsubdivisions can be realized geometrically so that the tiles stay"almost round" at all stages of the sequence. This problem isinteresting in its own right, but it is being studied here as part ofa deeper problem. It is a key feature of a program of theinvestigator and his collaborators to resolve the hyperbolic case ofWilliam P. Thurston's Geometrization Conjecture. The GeometrizationConjecture, which is the central outstanding problem inlow-dimensional topology (it includes the Poincare conjecture as aspecial case), states that every compact 3-manifold can be naturallysubdivided into geometric pieces. The techniques being developed forapproaching this have potential applications in other disciplines.