Date of Award

5-2022

Document Type

Thesis

Degree Name

Master of Arts in Mathematics

Department

Mathematics

First Reader/Committee Chair

Meyer, Jeffrey

Abstract

This thesis is centered around the construction and analysis of the principal arithmetic surface (3, 5) over Q. By adjoining the two symbols i,j, where i2 = 3, j2 = 5, such that ij = -ji, I can produce a quaternion algebra over Q. I use this quaternion algebra to find a discrete subgroup of SL2(R), which I identify with isometries of the hyperbolic plane. From this quaternion algebra, I produce a large list of matrices and apply them via Mobius transformations to the point (0, 2), which is the center of my Dirichlet domain. This list of transformed points is my orbit sampling. The possible walls of the Dirichlet domain are perpendicular bisectors of the hyperbolic geodesic segments between (0, 2) and points from my orbit sampling. Once I produced the list of all perpendicular bisectors, I plot them. From this plot, I refine the collection to only the walls for my Dirichlet domain. I then analyze this surface by finding its hyperbolic area, and the matrices that correspond to gluing one side to the other.

The production of this hyperbolic surface by adjoining (3,5) over Q is an “example problem”. This example problem produces a hyperbolic polygon that could be used to show others the process of finding a Dirichlet domain or producing a hyperbolic polygon. One reason why one may want to find a hyperbolic surface is for the list of generators that correlates to the edge gluings. This list of generators can be used in other areas of mathematics and science.

This project brought together many areas of math, including topology, hyperbolic geometry, linear algebra, group theory, and noncommutative algebra. Additionally, this project involved a significant programming component, in which I wrote and implemented code in Sage, a Python based computer algebra system.

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