Immersed Surfaces, Dehn Surgery and Essential Laminations

Citation

Li, Tao (2000) Immersed Surfaces, Dehn Surgery and Essential Laminations. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/4kag-zt09. https://resolver.caltech.edu/CaltechTHESIS:11212019-175145827

Abstract

Let M be an orientable and irreducible 3-manifold whose boundary is an incompressible torus. We are interested in immersed essential surfaces in closed 3-manifolds obtained from Dehn fillings on M. We show the following two things.

In Chapter 2, we suppose that M does not contain closed non-peripheral incompressible surfaces. We show that the immersed surfaces in M with the 4-plane property can realize only finitely many slopes. Moreover, we show that only finitely many Dehn fillings on M can yield 3-manifolds that admit non-positive cubing. This gives the first examples of hyperbolic 3-manifolds that cannot admit non-positive cubing.

In Chapter 3, we suppose M is hyperbolic. We show that all but finitely many Dehn fillings on M yield 3-manifolds that contain closed essential surfaces. Moreover, we give a bound on the number of exceptional Dehn fillings.

Thesis Files