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

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.