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.

最近在 3-流形几何和拓扑方面的大量成果为组合和几何地理解 3-流形提供了许多新工具。 特别是，如何获得对闭合双曲 3 流形的内部几何结构的明确理解的问题可以通过 P.I. 开发的新技术来解决。 与迪克·金丝雀（Dick Canary）和亚尔·明斯基（Yair Minsky）一起适用于无限体积的情况。 这种理解将是莫斯托刚性的一种有效版本，其中人们不仅知道双曲结构的唯一性，而且还知道其几何形状的明确描述。 P.I. 与 Juan Souto 合作，试图开发这种针对闭双曲 3 流形的图像，该流形承认 Heegaard 分裂，以 Heegaard 曲面的形式给出。此外，霍华德·马苏尔 (Howard Masur) 和亚伊尔·明斯基 (Yair Minsky) 还担任 P.I.将相当于一个表面的双曲 3 流形同伦的内部几何与 Teichmueller 空间中沿 Weil-Petersson 测地线 G 的表面几何联系起来。几何和拓扑研究的最新趋势是开发几何空间的组合模型。 这种对空间或形状的描述为了捕捉更多的大规模结构而牺牲了一定程度的精度，并且通常一般定理保证知道这种大规模结构就足以完全确定空间。 在 P.I. 最近的一项结果中R. Canary 和 Y. Minsky 的模型被用来对所有“不断负弯曲”或“双曲”的无限体积的三维空间进行分类，这些空间在某种意义上仍然是温和的。 这一结果解决了威廉·瑟斯顿长期以来的猜想，即某个数据（类似于空间的DNA序列）完全决定了其结构。 我们在有限体积情况下开发了类似的设置，并希望为此类空间证明类似的分类定理。 这种大规模类型的信息比几何结构的存在结果更有用，因为它可以让人们更全面地了解此类空间的行为方式。 如此大规模的数据在许多情况下都会出现并描述现象，无论是空间本身还是它们的参数空间。 P.I. 与他的合作者将继续开发所有双曲 3 维流形的大规模几何的完整描述，以及参数化它们的相关空间。