Trace classes in the modular and extended modular group

In this thesis, the trace classes in the modular group and the extended modular group are given. The relationship between these trace classes and block lenghts is explained. This thesis consists of five chapters, and in the first chapter, the studies on the subject are briefly mentioned. In the second chapter, the definitions, theorems, examples and results required for this study are given. In the third section, the elements of the modular group are classified according to their traces. The relationship between block length and trace is analyzed. A method that gives a representative set of trace classes for any trace is presented. In the fourth chapter, the trace classes in the extended modular groups are examined and the relationship between block lengths and trace is examined. In the fifth section, the results obtained from the topics discussed in the previous sections are briefly summarized.