New Relations for the Normal Subgroups of Hecke Groups

For an integer q ≥ 3, Hecke groups H(λq) are an important class of discrete groups withthe 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 area number of research papers on the properties of Hecke groups, their normal subgroups andthe relation with regular maps. Here we add some recent results to state new relations betweenthe parameters of the normal subgroups of Hecke groups and the corresponding regular mapswhich are also graphs in combinatorial sense by means of a new graph invariant called omegawhich was recently defined in 2018.