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

近年来在三维流形几何和拓扑方面的研究成果激增，为从组合和几何角度理解三维流形提供了许多新的工具。特别是，如何获得一个封闭的双曲3流形的内部几何的明确理解的问题，可以解决由P.I.与迪克金丝雀和yair Minsky一起开发的新技术，适用于无限体积的情况。这样的理解将是一种有效版本的Mostow刚性，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，试图发展承认Heegaard分裂的封闭双曲3流形的这种图像，给出Heegaard曲面。此外，与HowardMasur和Yair Minsky一起，P.I.将把相当于曲面的双曲3流形同伦的内部几何与沿Weil-Petersson测地线G inTeichmueller空间的曲面几何联系起来。几何和拓扑学研究的一个最新趋势是发展几何空间的组合模型。这种对空间或形状的描述牺牲了一定程度的精度，以捕获更多的大尺度结构，并且通常一般定理保证知道这个大尺度结构足以完全确定空间。在最近与R. Canary和Y. Minsky合作的P.I.中，这些模型被用来对所有“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是弯曲的。这一结果解决了威廉·瑟斯顿长期以来的猜想，即某一特定数据（类似于空间的某种dna序列）完全决定了它的结构。我们已经在有限体积的情况下建立了一个类似的设置，并希望为这样的空间证明一个类似的分类定理。这种大尺度类型的信息比几何结构的存在性结果更有用，因为它可以让人们更完整地了解这些空间的行为方式。这种大规模的数据出现并描述了许多情况下的现象，无论是空间本身，还是它们的参数空间。p.i.将与他的合作者一起，继续发展所有双曲三维流形的大尺度几何的完整描述，以及参数化它们的相关空间。

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.