Please use this identifier to cite or link to this item: http://arks.princeton.edu/ark:/88435/dsp01vq27zn569
 Title: On Closed Hyperbolic 3-manifolds and Pseudo-Anosov Maps Authors: Sun, Hongbin Advisors: Gabai, David Contributors: Mathematics Department Keywords: 3 manifoldfinite coverhyperbolic geometrypseudo-Anosov maps Subjects: Mathematics Issue Date: 2014 Publisher: Princeton, NJ : Princeton University 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). URI: http://arks.princeton.edu/ark:/88435/dsp01vq27zn569 Alternate format: The Mudd Manuscript Library retains one bound copy of each dissertation. Search for these copies in the library's main catalog Type of Material: Academic dissertations (Ph.D.) Language: en Appears in Collections: Mathematics