Regular maps and normal subgroups of hecke groups

In this PhD thesis, the normal subgroups of Hecke groups which were defined by Erich Hecke in an article published in 1936 were determined with the help of a one-to-one correspondence between regular maps. This correspondence was described by Jones and Singerman in 1978. Since regular maps up to genus 7 were known until 1993, the signatures of normal subgroups of Hecke groups up to genus 7 were determined by Cangül, and their group structures and characteristics were studied. In 2001, Conder and Dobcsanyi classified the orientable regular maps up to genus 15. It was possible to study normal subgroups of Hecke groups up to genus 101 in 2006, with Conder's classification of orientable regular maps up to genus 101. In 2011, Conder classified all orientable regular maps up to genus 303. In addition, a new invariant, which is called Omega invariant, defined by Delen and Cangül in 2018, was defined and the properties of this invariant were mentioned and relations were determined between the rank of the normal subgroups of Hecke groups and some graph parameters. The thesis consists of five chapters. In the first chapter, a brief information about the thesis is given and the definitions, basic concepts and theorems used in the formation of the thesis are given. In the second part, the historical theories that constitute the source of the thesis, and in the third part, the mathematical theories, theorems and methods used to reach the results obtained in the thesis are mentioned. The fourth chapter is the main part of the thesis. New findings, definitions, theorems and inferences about the thesis are given in this section. In the fifth and last part, the discussion and conclusions about the thesis are mentioned, and it is mentioned that the new findings that are the source of the writing of the thesis can be used in other fields.