On Closed Hyperbolic 3-manifolds and Pseudo-Anosov Maps

Abstract

This dissertation consists of two different research topics.

The first topic is a study on virtual properties of closed hyperbolic 3-manifolds. By applying Kahn-Markovic's and Liu-Markovic's construction of immersed almost totally geodesic surfaces in closed hyperbolic 3-manifolds, we construct various interesting immersed $\pi_1$-injective 2-complexes in closed hyperbolic 3-manifolds. By using these immersed $\pi_1$-injective 2-complexes and Agol's result that the groups of hyperbolic 3-manifolds are LERF, we show two results on virtual properties of closed hyperbolic 3-manifolds. The first results is, any finite abelian group is a direct summand of the virtual homology of any closed hyperbolic 3-manifold. The second result is, any closed oriented hyperbolic 3-manifold virtually 2-dominates any closed oriented 3-manifold.

The second topic is a study of pseudo-Anosov maps by using 3-manifold topology. For a hyperbolic surface bundle over the circle, we study the dilatation function defined on Thurston's fibered cone containing the given fibered structure. By using coordinates of the minimal point of the restriction of this dilatation function on the fibered face, we define an invariant of pseudo-Anosov maps, which is a $\mathbb{Q}$-submodule of $\mathbb{R}$. We will develop a few nice properties of this invariant, and give a few examples to show that this invariant can be nontrivial, i.e. the minimal point need not be a rational point (actually transcendental in this case).