This course served as an introduction to the many ways in which hyperbolic geometry has affected various parts of mathematics.
In the first part of the course, following Stahls' "The Poincare Half-Plane", we investigated the origins of the subject. This included lectures on the geometry of the hyperbolic plane. We studied the differences between Euclidean and Hyperbolic geometry. Areas of triangles, sums of angles, and isometries were all investigated.
After studying isometries of the upper half plane we were ready to deal with covering spaces and show that the hyperbolic plane is the universal cover of closed orientable 2 manifolds with positive genus. This served as an introduction to some important areas of topology.
The third area we investigated was the notion of a hyperbolic group (in the sense of Gromov). Triangles on the upper half plane are delta thin. This is easily seen to be a property of metric spaces and it follows that it is well defined for the Cayley graph of a group. A group is said to be Hyperbolic if its Cayley graph is delta thin. Hyperbolic groups include the fundamental group of a surface.
Aside from learning some beatiful mathematics we saw how ideas from one area of mathematics can affect other areas as well.
References: