Dissertations, Theses, and Capstone Projects
Date of Degree
6-2016
Document Type
Dissertation
Degree Name
Ph.D.
Program
Mathematics
Advisor
Abhijit Champanerkar
Committee Members
Ara Basmajian
Abhijit Champanerkar
Ilya Kofman
John Voight
Subject Categories
Geometry and Topology
Keywords
Macfarlane, arithmetic invariants, Dirichlet domains, hyperbolic surfaces, quaternion algebras, hyperbolic 3-manifolds
Abstract
I use a classical idea of Macfarlane to obtain a complex quaternion model for hyperbolic 3-space and its group of orientation-preserving isometries, analogous to Hamilton’s famous result on Euclidean rotations. I generalize this to quaternion models over number fields for the action of Kleinian groups on hyperbolic 3-space, using arithmetic invariants of the corresponding hyperbolic 3-manifolds. The class of manifolds to which this technique applies includes all cusped arithmetic manifolds and infinitely many commensurability classes of cusped non-arithmetic, compact arithmetic, and compact non-arithmetic manifolds. I obtain analogous results for actions of Fuchsian groups on the hyperbolic plane. I develop new tools to study such manifolds, and then focus on a new algorithm for computing their Dirichlet domains.
Recommended Citation
Quinn, Joseph, "Quaternion Algebras and Hyperbolic 3-Manifolds" (2016). CUNY Academic Works.
https://academicworks.cuny.edu/gc_etds/1354