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

近年来，三维流形的几何和拓扑研究取得了一系列新的成果，为从组合和几何的角度理解三维流形提供了新的工具。 特别是，如何获得一个封闭的双曲三维流形的内部几何的明确理解的问题，可以解决与新技术开发的P.I. 与Dick Canary和Yair Minsky一起适用于无限体积的情况。 这样的理解是Mostow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，而且还知道它的几何形状的明确描述。 私家侦探，injoint工作与胡安Souto，旨在发展这种图片forclosed双曲3-流形承认一个Heegaard分裂，给出的Heegaard表面。此外，与HowardMasur和Yair Minsky的P.I.将把等价于曲面的双曲3-流形的内部几何与Teichmueller空间中沿着Weil-Petersson测地线G的曲面几何联系起来。 这种对空间或形状的描述为了捕捉更多的大尺度结构而牺牲了一定程度的精确度，并且通常一般定理保证知道这种大尺度结构足以完全确定空间。 在P.I.与R. Canary和Y.明斯基，这样的模型被用来分类所有的“不断消极弯曲，”或“双曲线”的三维空间的无限体积，但驯服在一定意义上。 这一结果解决了威廉瑟斯顿的一个长期存在的猜想，即在某段数据中（类似于空间的一种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，并希望证明这样的空间类似的分类定理。 这种大尺度类型的信息比几何结构的存在性结果更有用，因为它给出了一个更完整的关于这种空间如何表现的图像。 这种大规模的数据出现并描述了许多背景下的现象，无论是空间本身，还是它们的参数空间。 私家侦探，与他的合作者一起，将继续发展一个完整的描述所有双曲三维流形的大规模几何，以及为相关的空间，参数化他们。

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.