Modular groups and planar maps

TL;DR

This paper establishes a correspondence between certain subgroups of the modular group and trivalent maps on a sphere, providing explicit formulas for counting and classifying these subgroups, especially those of index 18 with notable geometric features.

Contribution

It introduces a novel explicit formula linking genus zero, torsion-free subgroups of the modular group to planar maps, enhancing understanding of their structure and classification.

Findings

Derived formulas for subgroup counts and conjugacy classes

Identified special properties of index 18 subgroups

Established a correspondence with trivalent planar maps

Abstract

In this paper 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.