Virtual properties of hyperbolic 3-manifolds

三维流形局部看起来像三维空间。经典物理学将我们生活的宇宙描述为一个三流形。双曲几何是一种（非欧几里得）几何模型，满足欧几里得的前四个公设，但不满足第五个（平行公设）。对于3流形，双曲几何比欧几里德几何重要得多。Thurston和Perelman的作品暗示“大多数”3-流形具有双曲几何结构（即它们是双曲3-流形）。这些双曲3流形可以用所谓的2乘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。虽然矩阵群的子群看起来很容易理解，但双曲3-流形覆盖空间的各种性质实际上是相当神秘的。PI计划通过各种几何和代数方法研究双曲型3流形的覆盖空间。本课题的主要目的是研究双曲3-流形的有限覆盖，并利用纤维双曲3-流形的有限覆盖来研究伪anosov映射的有限覆盖。该研究将基于在虚拟Haken和虚拟光纤猜想的最新进展中开发的工具。特别值得一提的是，PI将重点关注三个主题。第一个主题是关于3-流形有限覆盖的拓扑不变量的渐近行为（例如，Seifert体积，同调扭转的大小）。第二个课题是利用双曲3流形的有限覆盖来研究伪anosov映射的有限覆盖（如伪anosov映射的虚同调谱半径）。第三个课题是研究小膨胀伪anosov映射的映射环面（例如，找到生成所有最小膨胀伪anosov映射的双曲3-流形的显式有限集合）。从双曲几何，低维拓扑，几何群论，动力系统和度量几何方法将涉及到工作。

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.