Introduction to hyperbolic geometry

Abstract

The main purpose of this scientific initiation project is to develop the mathematical maturity of the student and to stimulate his curiosity and his scientific interest. The theme chosen to achieve this purpose was Hyperbolic Geometry because it allows an interesting interconnection between General Topology concepts and Differential Geometry (studied by the student in the last year, in scientific initiation with the same tutor), with the courses of Abstract Algebra (courses in the last two years of undergraduate) and Complex Analysis that he will take this semester (2019-1). The final objective of this project is the construction of examples of abstract hyperbolic surfaces from the action of discrete groups of isometries of the plane hyperbolic (Fuchsian Groups). More details are presented in the section entitled "Work Plan." It is worth noting that this project does not aim to have content equivalent to that of a complete course of Hyperbolic Geometry. However, we are convinced that it will contribute to a better use of the student in the disciplines of his undergraduate course and, in the future, to graduate.