Date of Award

May 2019

Degree Type

Dissertation

Degree Name

Doctor of Philosophy

Department

Mathematics

First Advisor

Chris Hruska

Committee Members

Craig Guilbault, Boris Okun, Burns Healy, Jeb Willenbring

Abstract

A compact, orientable, irreducible 3-manifold M with empty or toroidal boundary is

called geometric if its interior admits a geometric structure in the sense of Thurston. The

manifold M is called non-geometric if it is not geometric. Coarse geometry of an immersed

surface in a geometric 3-manifold is relatively well-understood by previous work of Hass,

Bonahon-Thurston. In this dissertation, we study the coarse geometry of an immersed

surface in a non-geometric 3- manifold.

The first chapter of this dissertation is a joint work with my advisor, Chris Hruska. We

answer a question of Daniel Wise about distortion of a horizontal surface subgroup in a graph

manifold. We show that the surface subgroup is quadratically distorted in the fundamental

group of the graph manifold whenever the surface is virtually embedded (i.e., separable) and

is exponentially distorted when the surface is not virtually embedded.

The second chapter of this dissertation generalizes the previous work of the author and

Hruska to surface subgroups in non-geometric 3-manifold groups. We show that the only

possibility of the distortion is linear, quadratic, exponential, and double exponential. We

also establish a strong connection between the distortion and the separability of surface

subgroups in non-geometric 3-manifold groups.

The final chapter of the dissertation makes a progress in understanding the structure of

the group of quasi-isometries of a closed graph manifold which is mysterious.

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