My aim in this note is to give a brief survey of one particular aspect of the modular group r “ PSL2(Z), namely the balance (or rather the lack of it) between its congruence and non-congruence subgroups. Among the arithmetic subgroups (those of finite index) in r, the congruence subgroups have proved to be the most important and the most widely studied. Nevertheless, it has been known for some time that, in a certain sense,
“most of the arithmetic subgroups of Γ are
I shall describe several recent lines of investigation which, in different ways, add substance to this rather tenuous statement.
The modular group (together with the closely related groups SL2(ℤ), GL2(ℤ) and PGL2(ℤ)) is like an octopus, with tentacles reaching out into many branches of pure mathematics; a complete survey is out of the question here, but I hope that, at the very least, the bibliography will enable the reader to discover more about this fascinating group and its applications. For background reading, the classic reference is still Klein and Fricke more modern treatments of various aspects of r can be found in, Fine surveys the similarities and differences between r and the Picard group PSL2(ℤ[i]), while other classes of groups closely related to r are considered in.
Let A denote the set of arithmetic subgroups of Γ, that is, those of finite index.