I - Hyperbolic Geometry

Summary

Summary. The aim of these lectures is to provide a self-contained introduction into the geometry of the hyperbolic plane and to develop computational tools for the construction and the study of discrete groups of non-Euclidean motions.

The first lecture provides a minimal background in the classical geometries: spherical, hyperbolic and Euclidean. In Lecture 2 we present the Poincaré model of the hyperbolic plane. This is the most widely used model in the literature, but possibly not always the most suitable one for effective computation, and so we present in Lectures 4 and 5 an independent approach based on what we call the matrix model. Between the two approaches, in Lecture 3, we introduce the concepts of a Fuchsian group, its fundamental domains and the hyperbolic surfaces. In the final lecture we bring everything together and study, as a special subject, the construction of hyperbolic surfaces of genus 2 based on geodesic octagons.

The course has been designed such that the reader may work himself linearly through it. The prerequisites in differential geometry are kept to a minimum and are largely covered, for example by the first chapters of the books by Do Carmo [8] or Lee [15]. The numerous exercises in the text are all of a computational rather than a problem-to-solve nature and are, hopefully, not too hard to do.