Date of Degree

9-2019

Document Type

Dissertation

Degree Name

Ph.D.

Program

Mathematics

Advisor

Abhijit Champanerkar

Advisor

Dennis Sullivan

Committee Members

Ilya Kofman

Abstract

The Geometrization Theorem for 3-manifolds states that every closed orientable 3-manifold can be cut along spheres and tori into pieces which have a geometric structure modeled on one of the eight, 3-dimensional geometries. In joint work with Dennis Sullivan, we combine the different geometries on the toroidal ends of 3-manifolds to describe a uniform geometric structure for all oriented closed prime 3-manifolds. Hyperbolic structures on links in the thickened torus and their geometric properties have been of great interest recently. We discuss geometric properties of augmented and fully augmented links in the thickened torus. We show how sequences of fully augmented links in the 3-sphere which diagrammatically converge to a biperiodic fully augmented link have interesting asymptotic volume growth.

Share

COinS