Large Scale Geometry of Surfaces in 3-Manifolds

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.