Date of Award
Doctor of Philosophy
Craig Guilbault, Boris Okun, Burns Healy, Jeb Willenbring
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.
Nguyen, Hoang Thanh, "Large Scale Geometry of Surfaces in 3-Manifolds" (2019). Theses and Dissertations. 2106.