Normal subgroups of Hecke groups on sphere and torus

Abstract

We use regular map theory to obtain all normal subgroups of Hecke groups of genus 0 and 1. The existence of a regular map corresponding uniquely to every normal subgroup of Hecke groups $H(lambda_q)$ is a result of Jones and Singerman, and it is frequently used here to obtain normal subgroups. It is found that when q is even, $H(lambda_q)$ has infinitely many normal subgroups on the sphere, while for odd q, this number is finite. The total number of normal subgroups of $H(lambda_q)$ on a torus is found to be either 0 or infinite. The latter case appears iff q is a multiple of 4. Finally, a result of Rosenberger and Kern-Isberner is reproved here.