Formulas and multiplicative relationships for the parameters of subgroups of the modular group

Abstract

I. Introduction Let F = PSL(2, Z) be the classical modular group. Let G be a subgroup off of finite index. The most important parameters associated with G are/~ =/~(G), the index of G; t = t(G), the number of parabolic classes of G; e2 = e2(G), the number of elliptic classes of order 2 of G; e3 = e3 (G), the number of elliptic classes of order 3 of G; and g=g(G), the genus of G. These are not independent but are related by the formula bt t e 2 e 3 g= 1 + 12 2 4 3 ' (1) which is just the hyperbolic area formula. The point of view we take here is the following: we assume that a left coset decomposition for F modulo G is known. From this decomposition, we determine the parameters described above. We then apply these results to the case of congruence groups, and prove that/~, t, e2, e3 are multiplicative arithmetic functions (in a well-defined sense) of the level. Finally, we mention briefly some applications to the determination of congruence groups with various restrictions imposed on the parameters; for example, with a fixed number of parabolic classes. If f2 = SL(2, Z) and ~0 is the natural homomorphism of t2 modulo its center C = {I, - I}, then