Immersing Essential Surfaces in Odd Dimensional Closed Hyperbolic Manifolds

• Main Author: Fu, Lei
• Format: Others
• Language: en
• Published: 2017
• Online Access: https://thesis.library.caltech.edu/10298/1/Fu_Lei-main.pdf; 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 <https://resolver.caltech.edu/CaltechThesis:06062017-094601489>

<p>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.</p> <p>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.</p>