A panaroma of the fundamental group of the modular orbifold

TL;DR

This paper provides a comprehensive overview of the subgroup structure of the modular group, including both finite index and infinite index subgroups, and explores related arithmetic questions.

Contribution

It offers a unified perspective on the subgroup categories of the modular group and discusses arithmetic implications in both tame and non-tame parts.

Findings

Classification of subgroups into tame and non-tame categories

Discussion of arithmetic questions related to Belyi's theorem

Insights into the structure of the modular orbifold's fundamental group

Abstract

We give an overview of the category of subgroups of the modular group, incorporating both the tame part, i.e. finite index subgroups, and the non-tame part, i.e. the rest. We also discuss arithmetic related questions which exist in both the tame part (via Belyi's theorem) and the non-tame part.