Modular Group - Definition

Definition

The modular group Γ is the group of linear fractional transformations of the upper half of the complex plane which have the form

where a, b, c, and d are integers, and ad − bc = 1. The group operation is function composition.

This group of transformations is isomorphic to the projective special linear group PSL(2,Z), which is the quotient of the 2-dimensional special linear group SL(2,Z) over the integers by its center {I, −I}. In other words, PSL(2,Z) consists of all matrices

where a, b, c, and d are integers, ad − bc = 1, and pairs of matrices A and −A are considered to be identical. The group operation is the usual multiplication of matrices.

Some authors define the modular group to be PSL(2,Z), and still others define the modular group to be the larger group SL(2,Z). However, even those who define the modular group to be PSL(2,Z) use the notation of SL(2,Z), with the understanding that matrices are only determined up to sign.

Some mathematical relations require the consideration of the group S*L(2,Z) of matrices with determinant plus or minus one. (SL(2, Z) is a subgroup of this group.) Similarly, PS*L(2,Z) is the quotient group S*L(2,Z)/{I, −I}. A 2x2 matrix with unit determinant is a symplectic matrix, and thus SL(2,Z)=Sp(2,Z), the symplectic group of 2x2 matrices.

One can also use the notation GL(2,Z) for S*L(2,Z), because an integer matrix is invertible if and only if it has determinant equal to ±1. Alternatively, one may use the explicit notation SL±(2,Z).