Teichmüller Theory, Kleinian Groups, and the Complex of Curves

泰希米勒理论、克莱尼群和曲线复形

• 批准号: 0906229
• 负责人: Jeffrey Brock
• 金额: $ 21.08万
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-01 至 2013-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0906229&HistoricalAwards=false
• 关键词: Teichm; ller; Theory; Kleinian; Groups

The classification of finitely generated Kleinian groups and their associated quotient hyperbolic 3-manifolds has generated significant new tools for studying many problems at the interfaces of Teichmüller theory, Kleinian groups and low-dimensional topology. In particular, the existence of a model manifold for a hyperbolic 3-manifold M that is uniformly bi-Lipschitz to M has foregrounded the extent to which many problems in the study of closed and finite volume hyperbolic 3- manifolds can be understood in terms of combinatorial structures associated to surfaces. Likewise, large-scale questions in the geometry of Teichmüller space have come into relief in terms of a new understanding of these combinatorics: the asymptotic geometry of geodesics in various metrics has been reconstituted and understood in a new language, yet the structure of the classical Weil-Petersson metric from this point of view remains largely unclear and tantalizingly open. Our proposed research will demonstrate how model manifolds serve as building blocks for hyperbolic structures on closed manifolds via Heegaard splittings, to develop control on the synthetic geometry and dynamics of the Weil-Petersson metric on Teichmüller space via the complex of curves, and to continue to reveal applications of the model manifolds to the topology of deformation spaces of hyperbolic 3-manifolds.The idea of a "coarse model" in geometry proposes that one might sacrifice a certain degree of precision in the interest of capturing more large-scale structure. Frequently a coarse model plays a similar role to DNA in biology: it can determine fine features of a space despite its apparently coarse nature. In a recent result of the P.I. with R. Canary and Y. Minsky, such models were used to classify all `constantly negatively curved,' or `hyperbolic' three-dimensional spaces of infinite volume that are `tame' in a certain sense. The classification result solved a long-standing conjecture of William Thurston, and opened the door to developing a more detailed and complete picture of geometries on manifolds previously considered understood. After Perelman's solution to Thurston's geometrization conjecture and the famous Poincaré conjecture, the groundwork is in place for a fundamental investigation of algebraic, geometric and topological properties of all spaces of 3-dimensions and how these properties interrelate.

对生成的Kleinian群及其商双曲3-流形的分类为研究Teichmüller理论、Kleinian群和低维拓扑的接口上的许多问题提供了重要的新工具。 特别是，存在一个模型流形的双曲3-流形M，是一致的bi-Lipschitz到M已经前景的程度，在研究封闭和有限体积双曲3-流形的许多问题可以理解的组合结构方面的表面。 同样地，泰希缪勒空间几何中的大规模问题在对这些组合数学的新理解中得到了缓解：各种度量中测地线的渐近几何已经用一种新的语言重新构建和理解，然而从这个角度来看，经典威尔-彼得森度量的结构在很大程度上仍然不清楚，而且是开放的。 我们提出的研究将演示模型流形如何通过Heegaard分裂作为封闭流形上双曲结构的构建块，通过曲线的复杂性来控制Teichmüller空间上Weil-Petersson度量的合成几何和动力学，并继续揭示模型流形在双曲三维流形变形空间拓扑中的应用。在几何学中，提出为了捕捉更大规模的结构，可能会牺牲一定程度的精度。 通常，粗糙的模型扮演着类似于生物学中DNA的角色：它可以确定空间的精细特征，尽管它显然是粗糙的。 在PI最近的一项结果中 与R. Canary和Y.明斯基，这样的模型被用来分类所有的“不断负弯曲，”或“双曲”的三维空间的无限体积是“驯服”在某种意义上。 分类结果解决了威廉·瑟斯顿的一个长期猜想，并打开了一扇大门，以发展一个更详细和完整的图像几何流形以前认为理解。 在佩雷尔曼解决瑟斯顿的几何化猜想和著名的庞加莱猜想之后，对所有三维空间的代数、几何和拓扑性质以及这些性质如何相互关联进行了根本性的研究。