The Groups Acting On the Riemann Sphere

Ruba Yousef Wadi


In this Master thesis we consider the group actions, with emphasis on the group of general Möbius transformation of one complex valuable acting on the Riemann sphere.

We study some invariant subspaces of Riemann sphere under the actions of natural groups of transformations, including the invariant quantities in Hyperbolic Geometry that is a beautiful area of Mathematics.

We use analytic and algebraic points of view to describe some group actions on Riemann sphere; in particular, we present the relationships between isometries of hyperbolic plane, Möbius transformations, and groups of matrices.