On the elements of extended modular group that fix rational numbers

In this study, the elements of the extended modular group that fix the rational numbers are examined. For this purpose, properties of continued fractions, the Farey sequence, and the Farey graph are used. Every rational number has a finite continued fraction expansion. The successive convergences of this continued fraction expansion are the vertices of a path from infinity to the rational number in the Farey graph. This study consists of five chapters. In the first chapter, which is the introduction, the study is briefly introduced. In the second chapter, the basic concepts and theorems that are related to the subject of the study are given. In the third chapter, the results obtained from the studies about the subject are given. The fourth chapter is the main part of the thesis. In this section, the results are given with their proofs. First, with the help of Farey neighbors, matrix representatives of parabolic and reflection elements that fix a given rational number are obtained. Then, using continued fractions and paths in the Farey graph, these elements are expressed as a word of generators. The block reduced forms of these words are also given in the same section. Finally, it is explained with examples of how the new block reduced form can be obtained as the product of matrices whose entries are Fibonacci numbers.