Effective Rigidity, Combinatorial Models, and Parameter Spaces for Low-Dimensional Hyperbolic Manifolds

低维双曲流形的有效刚性、组合模型和参数空间

• 批准号: 0505442
• 负责人: Jeffrey Brock
• 金额: --
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-01 至 2009-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0505442&HistoricalAwards=false
• 关键词: Effective; Rigidity; Combinatorial; Models; Parameter

The recent surge of results in the geometry and topology of3-manifolds has provided many new tools for understanding3-manifolds combinatorially and geometrically. In particular, thequestion of how to gain an explicit understanding of the internalgeometry of a closed hyperbolic 3-manifold can be addressed withnew techniques developed by the P.I. together with Dick Canary andYair Minsky that apply to the infinite volume case. Such anunderstanding would be a kind of effective version of Mostow rigidity,wherein one not only knows the uniqueness of the hyperbolic structurebut additionally an explicit decription of its geometry. The P.I., injoint work with Juan Souto, seeks to develop this kind of picture forclosed hyperbolic 3-manifolds admitting a Heegaard splitting,given in terms of the Heegaard surface. Additionally, with HowardMasur and Yair Minsky the P.I. will relate the internal geometry ofhyperbolic 3-manifolds homotopy equivalent to a surface to thegeometry of surfaces along a Weil-Petersson geodesic G inTeichmueller space.A recent trend in the study of geometry and topology is to developcombinatorial models for geometric spaces. This kind of descriptionof a space or shape sacrifices a certain degree of precision in theinterest of capturing more of the large-scale structure, and oftengeneral theorems guarantee that knowing this large-scale structure issufficient to completely determine the space. In a recent result ofthe P.I. with R. Canary and Y. Minsky, such models were used toclassify all `constantly negatively curved,' or `hyperbolic'3-dimensional spaces of infinite volume that are nevertheless tamein a certain sense. This result solved a long-standing conjecture ofWilliam thurston, where in a certain piece of data (akin to a kind ofDNA-sequence for the space) completely determines its structure. Wehave developed a similar setup in the finite-volume case, and hope toprove a similar classification theorem for such spaces. Informationof this large-scale type is more useful than existence results forgeometric structures, in that it gives one a more complete picture ofhow such spaces behave. Such large-scale data arise and describephenomena in many contexts, whether it is the spaces themselves, ortheir parameter spaces. The P.I., together with his collaborators,will continue to develop a complete description of the large-scalegeometry of all hyperbolic 3-dimensional manifolds, as well as forrelated spaces that parameterize them.

3-manifolds的几何形状和拓扑结果的最新结果提供了许多新的工具，可以从几何上和几何上理解3个manifolds。 特别是，如何通过P.I开发的新技术来解决如何获得对封闭双曲线3个manifold的内部几何测定法的明确理解的问题。 与迪克金丝雀Andyair Minsky一起，适用于无限的体积案例。 这样的无障碍将是一种有效的刚性版本，其中不仅知道双曲线结构的独特性，还知道其几何形状的明确降低。 P.I.与胡安·苏托（Juan Souto）一起工作，试图开发这种图像，以heegaard表面给出了heegaard拆分的悬停双曲线3型脉冲。此外，与霍华德马尔和耶尔·明斯基一起将与沿着Weil-Petersson Geodesic G Inteichmueller空间相当于表面的基因几何形状与表面的几何形状相关。 这种描述的空间或形状在捕获更多的大规模结构的问题上牺牲了一定程度的精确度，并且经常是tengeneral定理，确保知道这种大规模的结构足以完全确定空间。 在P.I.的最新结果中使用R.Canary和Y. Minsky，此类模型被用来分类所有“不断弯曲”或“无限量的超纤维” 3维空间，但仍然是某种意义上的tamein。 该结果解决了Wwilliam Thurston的长期猜想，其中在某些数据中（类似于该空间的DNA序列）完全决定了其结构。 Wehave在有限体积的情况下开发了类似的设置，并希望对此类空间进行类似的分类定理。 这种大规模类型的信息比存在结果较大的几何结构更有用，因为它使人们对这种空间的行为的表现更加完整。 出现这样的大规模数据并在许多情况下（无论是空间本身还是标题参数空间）中描述了ephenomena。 P.I.与他的合作者一起将继续对所有双曲线三维流形的大尺度测定法以及对它们进行参数化的相关空间的完整描述。