Subdivision Rules and 3-Manifold Topology

威廉·J·弗洛伊德这个项目试图解决瑟斯顿几何化猜想的双曲线化问题。具体地说，我们的目的是解决这样一个猜想，即一个在2-球面上具有无穷空间的Gromov-双曲群在双曲3-空间上有上紧的、适当的不连续作用。这位研究人员和他的合作者从证明Gromov-双曲群的无穷大空间上的某些递归平铺序列的一致性的角度来讨论这一猜想。以前的工作已经指出，递归平铺序列的一致性可能源于找到与递归结构相关联的分支表面的不变保形结构。这种可能性源于递归结构和有理映射之间的联系，而瑟斯顿关于2-球面的临界有限分支映射的分类定理给出了这一理论可能如何发展的洞察。关于扭面配对3-歧管的进一步工作也在计划中。扭面配对的许多基本理论已经完成，但一些问题仍然存在，这些问题是该理论进一步发展的核心。这里的重大进展可以帮助项目的主要部分，因为扭面配对是上述猜想的一个很好的测试样本来源。这项建议的直接焦点是平面配对的顺序。给定初始平铺和细分的组合规则，递归地获得初始平铺的细分序列。我们的目标是了解这些组合细分何时可以在几何上实现，从而使瓷砖在序列的所有阶段保持“几乎是圆形的”。这个问题本身很有趣，但这里将其作为更深层次问题的一部分进行研究。解决威廉·P·瑟斯顿的几何猜想的双曲型问题是这位研究人员和他的合作者计划的一个关键特点。几何猜想是低维拓扑学中最突出的问题(包括作为特例的Poincare猜想)，它指出每个紧致的3-流形都可以自然地细分为几何片。为达到这一目的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.