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.

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