Immersing Essential Surfaces in Odd Dimensional Closed Hyperbolic Manifolds

Citation

Fu, Lei (2017) Immersing Essential Surfaces in Odd Dimensional Closed Hyperbolic Manifolds. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/Z9XK8CKN. https://resolver.caltech.edu/CaltechThesis:06062017-094601489

Abstract

The surface subgroup theorem, proved by Kahn and Markovic, states that the fundamental group of every closed hyperbolic 3-manifold contains a closed hyperbolic surface subgroup. The criterion of incompressibility, a criterion to ensure that an immersing surface to be essential, has played an important role in their proof.

In this thesis, we generalize the criterion of incompressibility from dimension three to all higher dimensions. Then we use the mixing property of the geodesic flow to construct a closed immersed surface which satisfies the assumption of our criterion when the hyperbolic manifold is in an odd dimension. Together, we prove the surface subgroup theorem in all odd dimensions.

Thesis Files