We give an explicit formula for the number of subgroups of the modular group of a given index that are genus zero and torsion-free, and a formula for their conjugacy classes. We do so by exhibiting a correspondence between these groups and the trivalent maps on a sphere. We focus on the particular case of the subgroups of index 18 which have some interesting geometric properties.
"Modular groups and planar maps." Rocky Mountain J. Math. 51 (5) 1847 - 1863, October 2021. https://doi.org/10.1216/rmj.2021.51.1847