New relations for the normal subgroups of hecke groups

Abstract

For an integer q ≥ 3, Hecke groups H(λq) are an important class of discrete groups with the most important member being the famous modular group obtained in the case of q = 3. They were defined by E. Hecke in 1936 when he was studying with Dirichlet series. There are a number of research papers on the properties of Hecke groups, their normal subgroups and the relation with regular maps. Here we add some recent results to state new relations between the parameters of the normal subgroups of Hecke groups and the corresponding regular maps which are also graphs in combinatorial sense by means of a new graph invariant called omega which was recently defined in 2018.