Modular group

In mathematics, the modular group Γ (Gamma) = SL(2,Z) is the 2-dimensional special linear group over the integers. In other words, the modular group consists of all matrices

${\begin{pmatrix}a&b\\c&d\end{pmatrix}}$

where a, b, c, and d are integers, and ad - bc = 1. The operation is the usual multiplication of matrices. The modular group is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics.

Relationship to Hyperbolic Geometry

The modular group is important because it forms a subgroup of the group of isometries of the hyperbolic plane. If we consider the upper half-plane model H of hyperbolic plane geometry, then the group of all orientation-preserving isometries of H consists of all Möbius transformations of the form

$w={\frac {az+b}{cz+d}}$

where a, b, c, and d are real numbers and ad - bc = 1. Put differently, the group SL(2,R) acts on the upper half-plane H according to the following formula:

${\begin{pmatrix}a&b\\c&d\end{pmatrix}}\cdot z={\frac {az+b}{cz+d}}$

This (left-)action is not faithful because both the identity matrix I and its additive inverse -I fix all of H. For this reason, the group of orientation-preserving isometries of H is actually PSL(2,R), not SL(2,R). Similarly, the modular group does not act faithfully on H and many authors define the modular group to be PSL(2,Z) rather than SL(2,Z). The distinction is often glossed over in practice. In this article, the term modular group will refer to SL(2,Z), and whenever we wish to pass to the projective group, this will be made explicit by the common notation of raising a bar above Γ, i.e.

$\Gamma =SL(2,\mathbb {Z} ),\ {\overline {\Gamma }}=PSL(2,\mathbb {Z} )=SL(2,\mathbb {Z} )/\{\pm I\}$

Group-theoretic Properties

[...include here at least the expression of elements in terms of generators S and T...]

Applications to Number Theory

[...include here at least some mention of quadratic forms, the fundamental domain (modular curve) and modular forms...]

Congruence Subgroups

[...brief mention and definition of congruence subgroup, this really deserves its own article independent of Γ]