Teichmüller Theory, Kleinian Groups, and the Complex of Curves

有限生成Klein群及其相关的商双曲3-流形的分类为研究Teichmüler理论、Kleinan群和低维拓扑界面上的许多问题提供了重要的新工具。特别地，对M一致双Lipschitz的双曲三维流形M的模型流形的存在，突出了闭和有限体积双曲三维流形研究中的许多问题可以用与曲面相关的组合结构来理解的程度。同样，Teichmüler空间几何中的大规模问题也随着对这些组合学的新理解而浮现出来：各种度量中的测地线的渐近几何已经用一种新的语言重新构建和理解，但从这个角度来看，经典的Weil-Petersson度量的结构在很大程度上仍然不清楚，而且开放得令人着迷。我们的研究将展示模型流形如何通过Heegaard分裂作为闭流形上双曲结构的构建块，通过曲线的复数来发展对Teichmüler空间上的Weil-Petersson度量的综合几何和动力学的控制，并继续揭示模型流形在双曲三维流形的形变空间的拓扑中的应用。几何中的“粗模型”的想法提出，为了捕捉更大规模的结构，人们可能会牺牲一定的精度。通常，粗略模型在生物学中扮演着类似于DNA的角色：它可以确定一个空间的精细特征，尽管它看起来很粗糙。在P.I.与R.Canary和Y.Minsky最近的一项研究中，这种模型被用来对所有在某种意义上“温和”的无限体积的“常负曲线”或“双曲线”三维空间进行分类。分类结果解决了威廉·瑟斯顿长期以来的一个猜想，并为开发以前被认为已被理解的流形上更详细和完整的几何图景打开了大门。在Perelman解决了瑟斯顿的几何化猜想和著名的Poincaré猜想之后，为基础研究所有三维空间的代数、几何和拓扑性质以及这些性质是如何相互关联的奠定了基础。

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.