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

TL;DR

This work connects conjugacy classes of finite-index subgroups of $\mathrm{PSL}_2(\mathbb{Z})$ with bipartite cuboid graphs $Y_{\mathrm{comb}}(G)$ and provides an explicit, algorithmic pipeline to recover a convex fundamental domain (a special polygon) from the right action of $\Gamma$ on $G\setminus\Gamma$. It develops a complete combinatorial model and reduction framework: from $Y_{\mathrm{comb}}(G)$ to independent generators of $G$, to locating points in the fundamental domain, and to constructing explicit coset representatives for congruence subgroups, along with detailed automorphism-group analysis of modular curves. The approach yields significant computational improvements for congruence subgroups, achieving costs near the index rather than its square, and provides practical algorithms to construct the graph and polygon, compute representatives, and analyze automorphisms. These methods have broad implications for efficient computation of modular domains and for understanding the symmetries of modular curves via explicit, combinatorial data.

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 $Γ_0(N)$, $Γ_1(N)$, $Γ(N)$ 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 $\mathrm{SL}_2(\mathbb{Z}/N)/\{\pm I\}$.

Figures (10)

• Figure 1: Triangle $\triangle$.
• Figure 2: $Y_{\mathrm{comb}}(\Gamma)$.
• Figure 3: $Y_{\mathrm{comb}}(\Gamma(2))$.
• Figure 4: $Y_{\mathrm{comb}}(\Gamma_0(2))=Y_{\mathrm{comb}}(\Gamma^0(2))$ and distinguished edges, see p. \ref{['dist_edge_def']}.
• Figure 5: $Y_{\mathrm{comb}}(\ker(\Gamma\to{\mathbb Z}/2))$.

Theorems & Definitions (18)

• Theorem 2.1
• Proposition 2.2
• Theorem 3.1
• Theorem 3.2
• Lemma 5.1
• Proposition 5.2
• Proposition 5.3
• Proposition 5.4
• Proposition 5.5
• Lemma 5.6