Date of Award

5-2015

Document Type

Thesis

Degree Name

Master of Science (MS)

Department

Mathematical Sciences

First Advisor

Dr. Jianhua Gong

Second Advisor

Dr. Leonard Daus

Third Advisor

Dr. Ysukasa Yashiro

Abstract

In this Master thesis we consider the group actions, with emphasis on the group of general Möbius transformations 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. Keywords: Group actions, Riemann sphere, general Möbius transformations, transitivity, hyperbolic

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