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On computing finite index subgroups of PSL(2,Z)

Nicolás Mayorga Uruburu, Ariel Pacetti, Leandro Vendramin

TL;DR

The paper develops a recursive, computation-friendly framework to determine finite index subgroups of $PSL_2(\mathbb{Z})$ by encoding subgroups as bi-valent trees and Kulkarni diagrams, then deriving generalized Farey symbols and passports. The core advances include a structural analysis of bi-valent tree automorphisms, a practical algorithm to enumerate tree diagrams, and robust methods to recover subgroup data, cusps, and geometric invariants from these diagrams, all implemented in GAP/GAP4. It delivers extensive data for subgroups up to index $20$, distinguishes congruence from non-congruence cases, and demonstrates how the resulting modular curves’ arithmetic can be read from the combinatorial encodings. The work provides a publicly available computational tool and dataset to study the arithmetic of modular curves, including genus, cusp widths, and Hecke-operator related structure for congruence subgroups.

Abstract

We present a method to compute finite index subgroups of $PSL_2(\mathbb{Z})$. Our strategy follows Kulkarni's ideas, the main contribution being a recursive method to compute bivalent trees and their automorphism group. As a concrete application, we compute all subgroups of index up to 20. We then use this database to produce tables with several arithmetical properties.

On computing finite index subgroups of PSL(2,Z)

TL;DR

The paper develops a recursive, computation-friendly framework to determine finite index subgroups of by encoding subgroups as bi-valent trees and Kulkarni diagrams, then deriving generalized Farey symbols and passports. The core advances include a structural analysis of bi-valent tree automorphisms, a practical algorithm to enumerate tree diagrams, and robust methods to recover subgroup data, cusps, and geometric invariants from these diagrams, all implemented in GAP/GAP4. It delivers extensive data for subgroups up to index , distinguishes congruence from non-congruence cases, and demonstrates how the resulting modular curves’ arithmetic can be read from the combinatorial encodings. The work provides a publicly available computational tool and dataset to study the arithmetic of modular curves, including genus, cusp widths, and Hecke-operator related structure for congruence subgroups.

Abstract

We present a method to compute finite index subgroups of . Our strategy follows Kulkarni's ideas, the main contribution being a recursive method to compute bivalent trees and their automorphism group. As a concrete application, we compute all subgroups of index up to 20. We then use this database to produce tables with several arithmetical properties.
Paper Structure (14 sections, 26 theorems, 42 equations, 9 figures, 2 tables)

This paper contains 14 sections, 26 theorems, 42 equations, 9 figures, 2 tables.

Table of Contents

  1. Introduction
  2. A quick guide to Kulkarni's approach
  3. Bi-valent trees
  4. Automorphisms of bi-valent trees
  5. Algorithm to compute bi-valent trees
  6. Orientations on bi-valent trees
  7. Coloring external vertices
  8. Tree diagrams, generalized Farey symbols and subgroups
  9. Some important algorithms
  10. The word problem
  11. Relation with geometry
  12. Passports and equivalence classes of subgroups
  13. Congruence subgroups
  14. Some numerical data

Key Result

Theorem 2.4

A special polygon is a fundamental domain for the subgroup $\Delta$ generated by the side-pairing transformations and these transformations form an independent set of generators for $\Delta$. Conversely every subgroup $\Delta$ of finite index in ${\rm PSL}_2({\mathbb Z})$ admits a special polygon as

Figures (9)

  • Figure 1: Fundamental domain for ${\rm PGL}_2({\mathbb Z})$.
  • Figure 2: Unique tree in $\mathop{\mathrm{Bv}}\nolimits(2,3)$.
  • Figure 3: Unique oriented tree in $\mathop{\mathrm{Bv}}\nolimits(2,3)$ except isomorphism.
  • Figure 4: Relation between the distance and the crossed product.
  • Figure 5: Representatives near the infinity cusp.
  • ...and 4 more figures

Theorems & Definitions (82)

  • Definition 1.1
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Theorem 2.4: Kulkarni
  • proof
  • Definition 2.5
  • Example 2.6
  • Remark 2.7
  • Example 2.8
  • ...and 72 more