Virtual properties of hyperbolic 3-manifolds

三维流形在局部上看起来像三维空间。经典物理学将我们生活的宇宙描述为一个三维流形。双曲几何是一种(非欧几里得)几何模型，它满足欧几里得的前四个公设，但不满足第五个公设(平行公设)。事实证明，对于三维流形来说，双曲几何比欧几里得几何重要得多。瑟斯顿和佩雷尔曼的工作暗示“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)。这些双曲三维流形可以用二乘二矩阵的所谓离散群来描述，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，但双曲三维流形的覆盖空间的各种性质实际上是相当神秘的。PI计划用各种几何和代数方法来研究双曲三维流形的覆盖空间，这个项目的主要目的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。这项研究将基于在最近关于虚拟哈肯和虚拟纤维猜想的进展中开发的工具。特别是，国际和平研究所将重点关注三个主题。第一个主题是关于三维流形的有限覆盖的拓扑不变量(例如，Seifert体积、同调挠率的大小)的渐近行为。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映射的虚拟同调谱半径)。第三个主题是研究小伸缩伪Anosov映射的映射环面(例如，寻找生成所有最小伸缩伪Anosov映射的双曲3-流形的显式有限集合)。研究方法涉及双曲几何、低维拓扑学、几何群论、动力系统和度量几何。

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.