Normal subgroups of Hecke groups and regular maps

Abstract

Our main purpose is to explore the relationship between normal subgroups of Hecke groups and regular maps on compact orientable surfaces. We use regular maps to find all the normal subgroups of Hecke groups of index [les ]60. In Sections 1–4 we review the basic results concerning Hecke groups and normal subgroups of Fuchsian groups and in Section 5 we outline the basic facts we need about regular maps. The regular maps of genus 0 are the Platonic solids and in Section 6 we use these to completely determine the genus zero normal subgroups of Hecke groups. The regular maps of genus 1 were determined this century by Brahana [1]. We use them in Sections 7 and 8 to determine the genus 1 normal subgroups of Hecke groups. We give alternative proofs and extend theorems of M. Newman and also Kern-Isberner and Rosenberger. Unlike the genus 0 and 1 cases there are only finitely many regular maps of genus g[ges ]2. In Section 9 we use more recent results concerning their classification to find all normal subgroups of Hecke groups of index [les ]60. This was done for the modular group in [13].