Virtual properties of hyperbolic 3-manifolds

一个三维流形局部地看起来像三维空间。经典物理学把我们生活的宇宙描述为一个三维流形。双曲几何（英语：Hyperbolic geometry）是一种满足欧几里得前四个公设但不满足第五个公设（平行公设）的几何模型。事实证明，对于三维流形，双曲几何比欧几里得几何重要得多。Thurston和Perelman的工作暗示“大多数”三维流形具有双曲几何结构（即，它们是双曲三维流形）。这些双曲3-流形可以用所谓的2 × 2矩阵的离散群来描述，它们的覆盖空间对应于这些离散群的子群。虽然矩阵群的子群似乎很容易理解，但双曲三维流形的覆盖空间的各种性质实际上是非常神秘的。PI计划通过各种几何和代数方法研究双曲三维流形的覆盖空间，本项目的主要目标是研究双曲三维流形的有限覆盖，并利用纤维双曲三维流形的有限覆盖来研究伪Anosov映射的有限覆盖。这项研究将基于虚拟哈肯和虚拟纤维猜想的最新进展期间开发的工具。具体而言，PI将重点关注三个主题。第一个主题是关于3-流形的有限覆盖的拓扑不变量的渐近行为（例如，塞弗特体积，同调挠率的大小）。第二个主题是利用双曲三维流形的有限覆盖来研究伪Anosov映射的有限覆盖（例如，伪Anosov映射的虚同调谱半径）。第三个主题是研究小扩张伪Anosov映射的映射环面（例如，找到双曲3-流形的显式有限集合，生成所有最小膨胀的伪Anosov映射）。从双曲几何，低维拓扑，几何群论，动力系统，度量几何的方法将参与工作。

A 3-manifold looks like the 3-dimensional space locally. The classical physics describes the universe we live in as a 3-manifold. Hyperbolic geometry is a (non-Euclidean) geometric model satisfying Euclid's first four postulates but not the fifth (the parallel postulate). It turns out that hyperbolic geometry is much more important than Euclidean geometry for 3-manifolds. Thurston's and Perelman's works imply that "most" 3-manifolds have a hyperbolic geometric structure (i.e., they are hyperbolic 3-manifolds). These hyperbolic 3-manifolds can be described by so-called discrete groups of two-by-two matrices, and their covering spaces correspond to subgroups of these discrete groups. Although subgroups of the matrix groups may seem easy to understand, various properties of covering spaces of hyperbolic 3-manifolds are actually quite mysterious. The PI plans to investigate covering spaces of hyperbolic 3-manifolds by various geometric and algebraic methods.The main goal of this project is to investigate finite covers of hyperbolic 3-manifolds, and to use finite covers of fibered hyperbolic 3-manifolds to study finite covers of pseudo-Anosov maps. The research will be based on tools developed during the recent progress on the Virtual Haken and Virtual Fibered Conjectures. In particular, the PI will focus on three topics. The first topic is about the asymptotic behavior of topological invariants of finite covers of 3-manifolds (e.g., the Seifert volume, the size of homological torsion). The second topic is to use finite covers of hyperbolic 3-manifolds to study finite covers of pseudo-Anosov maps (e.g., the virtual homological spectral radii of pseudo-Anosov maps). The third topic is to study the mapping tori of small dilatation pseudo-Anosov maps (e.g., to find the explicit finite collection of hyperbolic 3-manifolds that generates all pseudo-Anosov maps of smallest dilatation). Methods from hyperbolic geometry, low-dimensional topology, geometric group theory, dynamical systems, and metric geometry will be involved in the work.