Congruence and Noncongruence Subgroups of Gamma(2) via Graphs on Surfaces

TL;DR

This paper explores the relationship between subgroups of Gamma(2) and bipartite graphs on surfaces, providing criteria and methods to identify and analyze congruence and noncongruence subgroups through graph-based techniques.

Contribution

It introduces elementary criteria for identifying noncongruence subgroups and methods to generate group generators from surface graphs, advancing the understanding of subgroup classification.

Findings

Elementary criteria for noncongruence subgroup identification

Method to produce generators from surface graphs

Procedure to determine congruence of a subgroup from a graph

Abstract

There is an established bijection between finite-index subgroups Gamma of Gamma(2) and bipartite graphs on surfaces, or, equivalently, certain triples of permutations. We utilize this relationship to study both congruence and noncongruence subgroups in terms of the corresponding graphs. We show some elementary criteria which can be used to identify many noncongruence subgroups. Given a graph on a surface, we have a method to produce generators for the corresponding group Gamma in terms of the generators of Gamma(2). Given generators for Gamma(2n), we show how to determine whether or not a graph of level 2n corresponds to a congruence subgroup.