Connected fundamental domains for congruence subgroups

TL;DR

This paper constructs canonical, connected fundamental domains for certain congruence subgroups of SL(2,Z) by analyzing the projective line over Z/NZ and introduces a new computable function related to multiplicities.

Contribution

It provides explicit connected fundamental domains for congruence subgroups and introduces a new function for analyzing the projective line over Z/NZ.

Findings

Fundamental domains for Γ₀(N), Γ₁(N), and Γ(N) are connected.

A new computable function W related to multiplicities is introduced.

Examples and visualizations of the fundamental domains are provided.

Abstract

We produce canonical sets of right coset representatives for the congruence subgroups $\Gamma_0(N)$, $\Gamma_1(N)$ and $\Gamma(N)$, and prove that the corresponding fundamental domains are connected. Key to our construction is a study of the projective line $P^1({\mathbb Z}/N{\mathbb Z})$ using a function $M: {\mathbb Z}/N{\mathbb Z}\to {\mathbb Z}_{\geq 0}$, representing multiplicities. We further study this function and show that it is simply one less than another much more computable function $W:{\mathbb Z}/N{\mathbb Z}\to {\mathbb N}$, of possible independent interest. We present some examples and pictures at the end.