VIRTUAL PROPERTIES OF HYPERBOLIC 3-MANIFOLDS

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.