Congruence subgroups of Hecke groups and regular dessins

Abstract

In this thesis we deal with dessins, that is tessellations of orientable surfaces, or from another point of view, two-cell embeddings of graphs on orientable surfaces. Our approach uses the connections of dessins with the Hecke groups Hq, and emphasizes the number-theoretic aspects of these connections. In Chapter 1 we deal with the modular group F, the simplest of the Hecke groups. We study the relations between the dessins associated with the principal congruence subgroups of F, the cosets of the special congruence subgroups of F, and the arithmetic of the finite ring Zjy. Our examples include well-known regular dessins as the icosahedron and the dessin {3, 7}s on Klein's surface of genus 3. In Chapter 2 we give some basic results on the Hecke groups and the closely related maximal real cyclotomic fields, concentrating on the factorization of the integers inside these fields. In Chapter 3 we extend the work of Chapter 1 to the other Hecke groups, especially the quadratic Hecke groups. The examples include regular dessins as the cube, the dodecahedron, the small stellated dodecahedron, and {4,5}6 on Bring's curve of genus 4. In Chapter 4 we find representations for the Hecke groups and their quotients by the principal congruence subgroups, and we use the results to do some necessary calculations. In Chapter 5, using the results of Chapter 1 as motivation, we reduce the problem of calculating the normaliser of certain subgroups of the Hecke groups into solving a system of congruences, and we solve the corresponding systems for F, H4, H6. Then, using another method we calculate the normaliser of these subgroups in PSZ^R) f°r the cases iJ4, H6, and we also calculate the corresponding quotients.

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