Modular group

In mathematics, the Modular group Γ = SL(2,Z) is the 2-dimensional projective 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, 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. 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 of hyperbolic plane geometry, the isometry group is given by 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 action is not faithful because both the identity matrix I and its additive inverse -I fix all of H. For this reason, the isometry group of H is actually PSL(2,R), not SL(2,R). Similarly, the modular group does not act faithfully 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.

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

It can be shown that every A ∈ Γ can be non-uniquely written as

$A=T^{n_{1}}ST^{n_{2}}S\dots ST^{n_{k}}$

for some k > 0 and coefficients n_i, where

$T={\begin{bmatrix}0&-1\\1&0\end{bmatrix}}$

and

$S={\begin{bmatrix}1&1\\0&1\end{bmatrix}}$

External links

• MathWorld: Modular group Γ