A KLEINIAN APPROACH TO FUNDAMENTAL REGIONS

<p>This thesis takes a Kleinian approach to hyperbolic geometry in order to illustrate the importance of discrete subgroups and their fundamental domains (fundamental regions). A brief history of Euclids Parallel Postulate and its relation to the discovery of hyperbolic geometry be given first. We will explore two models of hyperbolic $n$-space: Uⁿ and Bⁿ. Points, lines, distances, and spheres of these two models will be defined and examples in U², U³, and B² will be given. We will then discuss the isometries of Uⁿ and Bⁿ. These isometries, known as M\"obius transformations, have special properties and turn out to be linear fractional transformations when in U² and B². We will then study a bit of topology, specifically the topological groups relevant to the group of isometries of hyperbolic $n$-space, I(Hⁿ). Finally we will combine what we know about hyperbolic $n$-space and topological groups in order to study fundamental regions, fundamental domains, Dirichlet domains, and quotient spaces. Using examples in U², we will then illustrate how useful fundamental domains are when it comes to visualizing the geometry of quotient spaces.</p>

Name thesis:degree_name

Master of Arts in Mathematics

Level thesis:degree_level

Thesis

Discipline thesis:degree_discipline

Mathematics

Year dc:date.available

2014

Author dc:creator

• Hidalgo, Joshua L

Contributors dc:contributor

• Dr. Rolland Trapp

• hyperbolic
• geometry
• fundamental
• regions
• domain
• Algebraic Geometry
• Geometry and Topology

Repository record dc:identifier

https://scholarworks.lib.csusb.edu/etd/35

OAI identifier oai:identifier

oai:scholarworks.lib.csusb.edu:etd-1060