Date of Award

Spring 2021

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

First Advisor

Oh, Hee

Abstract

This thesis consists of five separate projects. They are organized into the following sections: 1. Orbit closures of unipotent flows for hyperbolic manifolds with Fuchsian ends. In joint work with Oh, we establish an analogue of Ratner's orbit closure theorem for any connected closed subgroup generated by unipotent elements in $\mathrm{SO}(d, 1)$ acting on the space $\Gamma\backslash\mathrm{SO}(d, 1)$, assuming that the associated hyperbolic manifold $M=\Gamma\backslash\mathbb H^d$ is a convex cocompact manifold with Fuchsian ends. 2. Topological proof of Benoist-Quint. Let $G=\mathrm{SO}^\circ(d,1)$, $\Delta

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