Rigidity, Volume, and Combinatorics in Hyperbolic Geometry

双曲几何中的刚度、体积和组合学

• 批准号: 1849892
• 负责人: Jeffrey Brock
• 金额: $ 16.22万
• 依托单位: Yale University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-07-01 至 2022-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1849892&HistoricalAwards=false
• 关键词: Rigidity; Volume; Combinatorics; Hyperbolic; Geometry

For much of the twentieth century, the challenge to describe the shape of three dimensional spaces was viewed as an algebraic problem: indeed it is through algebra that we first distinguish the (topological) structure of a sphere from that of a doughnut. Work of William Thurston brought the geometry of such spaces more clearly into view as a central feature to explore. The triumph of this approach was Perelman's proof of Thurston's geometrization conjecture, that each three-dimensional manifold could be naturally broken up into pieces, each with a uniform geometry. The study of these geometries is frequently reduced, via a notion of rigidity, to considering simple combinatorial structures arising from loops on surfaces. The proposed research will explore how these structures predict volume, diameter, length and other aspects of the geometry, relating to notions from quantum physics.A longstanding connection between volume of hyperbolic three-manifolds and Weil-Petersson distance relied on a combinatorial comparison via the pants graph which organizes maximal multicurves on a surface, yet other connections were known to arise from work of Witten via renormalized volume. Recent work of Schlenker made this connection explicit in the context of quasi-Fuchsian manifolds, and PI's proposed work will develop this idea further to explicitly relate fibered 3-manifolds and translation distiance. More generally, a primary project in the proposed research is to solidify and extend the connection between combinatorics and geometry in closed manifolds, and to continue to investigate the structure of deformation spaces of Kleinian groups with these tools. As an example of the power of these techniques, bi-Lipschitz models for 'random' Heegaard splittings provide a full solution (with Rivin and Souto) to the conjecture of Dunfield and Thurston that random Heegaard splittings are almost surely hyperbolic and their volume grows linearly. Finally, the PI will pursue projects with Minsky, and with Modami, Leininger and Rafi on the role of combinatorics in the geometry of geodesics in the Weil-Petersson metric, investigating disparities between Teichmuller and Weil-Petersson geodesics in terms of unique ergodicity.

在二十世纪的大部分时间里，描述三维空间形状的挑战被视为一个代数问题：事实上，正是通过代数，我们首先区分了球体和甜甜圈的（拓扑）结构。威廉·瑟斯顿的作品使这些空间的几何形状更加清晰地呈现在人们的视野中，作为探索的中心特征。这种方法的胜利是佩雷尔曼对瑟斯顿几何化猜想的证明，即每个三维流形都可以自然地分解成碎片，每个碎片都具有统一的几何形状。通过刚性概念，对这些几何形状的研究经常被简化为考虑由表面上的环产生的简单组合结构。拟议的研究将探索这些结构如何预测与量子物理学概念相关的体积、直径、长度和几何的其他方面。双曲三流形的体积和 Weil-Petersson 距离之间的长期联系依赖于通过以下方式进行的组合比较：裤子图在表面上组织最大的多曲线，但已知其他连接是由 Witten 通过重整体积的工作产生的。 Schlenker 最近的工作在准 Fuchsian 流形的背景下明确了这种联系，PI 提出的工作将进一步发展这一想法，以明确关联纤维 3 流形和平移距离。更一般地说，拟议研究的主要项目是巩固和扩展闭流形中组合数学和几何之间的联系，并继续使用这些工具研究克莱因群的变形空间结构。作为这些技术力量的一个例子，“随机”Heegaard 分裂的双 Lipschitz 模型为 Dunfield 和 Thurston 的猜想提供了完整的解决方案（与 Rivin 和 Souto 一起），即随机 Heegaard 分裂几乎肯定是双曲线的，并且它们的体积线性增长。最后，PI 将与 Minsky、Modami、Leininger 和 Rafi 一起开展项目，研究组合数学在 Weil-Petersson 度量测地线几何中的作用，研究 Teichmuller 和 Weil-Petersson 测地线在独特遍历性方面的差异。