Modular groups and planar maps
Abdellah Sebbar, Khalil Besrour

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.
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