Combinatorics of double cosets and fundamental domains for the subgroups of the modular group

TL;DR

This paper explores the combinatorial structure of subgroups of the modular group, providing algorithms to construct fundamental domains and analyze automorphisms of modular curves efficiently.

Contribution

It introduces a combinatorial method to recover bipartite cuboid graphs from subgroup actions, enabling efficient construction of fundamental domains and automorphism analysis.

Findings

Method for constructing fundamental domains with polynomial complexity in log N

Algorithms for locating points and expressing elements in the fundamental domain

Simplified proof of automorphism groups of modular curves

Abstract

As noticed by R.~Kulkarni, the conjugacy classes of subgroups of the modular group correspond bijectively to bipartite cuboid graphs. We'll explain how to recover the graph corresponding to a subgroup $G$ of $\mathrm{PSL}_2(\mathbb{Z})$ from the combinatorics of the right action of $\mathrm{PSL}_2(\mathbb{Z})$ on the right cosets $G\setminus\mathrm{PSL}_2(\mathbb{Z})$. This gives a method of constructing nice fundamental domains (which Kulkarni calls "special polygons") for the action of $G$ on the upper half plane. For the classical congruence subgroups $\Gamma_0(N)$, $\Gamma_1(N)$, $\Gamma(N)$ etc. the number of operations the method requires is the index times something that grows not faster than a polynomial in $\log N$. This is roughly the square root of the number of operations required by the naive procedure. We give algorithms to locate an element of the upper half-plane on the fundamental domain and to write a given element of $G$ as a product of independent generators. We also (re)prove a few related results about the automorphism groups of modular curves. For example, we give a simple proof that the automorphism group of $X(N)$ is $\mathrm{SL}_2(\mathbb{Z}/N)/\{\pm I\}$.