Lydia Wegerman Gemzöe: Introduction to Hyperbolic Geometry and Fuchsian Groups
Time: Thu 2021-08-26 09.00 - 10.00
Respondent: Lydia Wegerman Gemzöe
Abstract: This thesis is an introduction to hyperbolic geometry and Fuchsian groups. We will introduce the Poincaré models of the hyperbolic plane and give a matrix representation of the group of hyperbolic isometries. A Fuchsian group is a discrete group of orientation-preserving hyperbolic isometries. We will give a definition of a fundamental domain for a Fuchsian group and describe the relation between Fuchsian groups and hyperbolic tessellations. One of the main results of this work is the Poincaré Polygon Theorem, which states that given a hyperbolic polygon we can find, provided that certain conditions are met, a Fuchsian group which has this polygon as a fundamental domain.