Modular group - Definition and Overview

In mathematics, the modular group Γ (Gamma) is a group that is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics. The modular group can be represented as a group of geometric transformations or as a group of matrices.

Definition

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

<math>z\mapsto\frac{az+b}{cz+d}<math>

where a, b, c, and d are integers, and ad − bc = 1.

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

<math>\begin{pmatrix} a & b \\ c & d \end{pmatrix}<math>

where a, b, c, and d are integers, and 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.

Group-theoretic properties

The modular group can be shown to be generated by the two transformations

<math>S: z\mapsto -1/z<math>

<math>T: z\mapsto z+1<math>

so that every element in the modular group can be represented (in a non-unique way) by the composition of powers of S and T. Geometrically, S represents inversion in the unit circle followed by reflection about the line Re(z)=0, while T represents a unit translation to the right.

The generators S and T obey the relations S² = 1 and (ST)³ = 1. These are the only independent relations, so the modular group has the presentation:

<math>\Gamma \cong \langle S, T \mid S^2, (ST)^3 \rangle<math>

Using the generators S and ST instead of S and T, shows that the modular group is isomorphic to the free product of the cyclic groups C₂ and C₃:

<math>\Gamma \cong C_2 * C_3<math>

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

<math>z\mapsto \frac{az + b}{cz + d}<math>

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

<math>\begin{pmatrix} a & b \\ c & d \end{pmatrix}\cdot z \,= \,\frac{az + b}{cz + d}<math>

This (left-)action is faithful. Since PSL(2, Z) is a subgroup of PSL(2, R), the modular group is a subgroup of the group of orientation-preserving isometries of H.

Tessellation of the hyperbolic plane

A typical fundamental domain for the action of Γ on the upper half plane.

The modular group acts on H as a discrete subgroup of PSL(2, R), i.e. for each z in H we can find a neighbourhood of z which does not contain any other element of the orbit of z. This also means that we can construct fundamental domains, which (roughly) contain exactly one representative from the orbit of every z in H. (Care is needed on the boundary of the domain.)

There are many ways of constructing a fundamental domain, but a common choice is the region

<math>R = \left\{ z \in H: \left| z \right| > 1,\, \left| \,\mbox{Re}(z) \,\right| < \frac{1}{2} \right\}<math>

bounded by the vertical lines Re(z) = 1/2 and Re(z) = −1/2, and the circle |z| = 1. This region is a hyperbolic triangle. It has vertices at (1 + i√3)/2 and (−1 + i√3)/2, where the angle between its sides is π/3, and a third vertex at infinity, where the angle between its sides is 0.

By transforming this region in turn by each of the elements of the modular group, a regular tessellation of the hyperbolic plane by congruent hyperbolic triangles is created.

Congruence subgroups

Important subgroups of the modular group Γ, called congruence subgroups, are given by imposing congruence relations on the associated matrices.

There is a natural homomorphism SL(2,Z) → SL(2,Z_N) given by readings the entries modulo N. This induces a homomorphism on the modular group PSL(2,Z) → PSL(2,Z_N). The kernel of this homomorphism is called the principal congruence subgroup of level N, denoted Γ(N). We have the following short exact sequence:

<math>1\to\Gamma(N)\to\Gamma\to\mbox{PSL}(2,\mathbb{Z}_N)\to 1<math>.

Being the kernel of a homomorphism Γ(N) is a normal subgroup of the modular group Γ. The group Γ(N) is given as the set of all modular transformations

<math>z\mapsto\frac{az+b}{cz+d}<math>

for which a ≡ d ≡ ±1 (mod N) and b ≡ c ≡ 0 (mod N).

The principal congruence subgroup of level 2, Γ(2), is also called the modular group Λ. Since PSL(2,Z₂) is isomorphic to S₃, Λ is a subgroup of index 6. The group Λ consists of all modular transformations for which a and d are odd and b and c are even.

Another important family of congruence subgroups are the groups Γ₀(N) defined as the set of all modular transformations for which c ≡ 0 (mod N). Note that Γ(N) is a subgroup of Γ₀(N).

History

The modular group and its subgroups were first studied in detail by Dedekind and by Felix Klein as part of his Erlangen programme in the 1870s. However, the closely related elliptic functions were studied by Lagrange in 1785, and further results on elliptic functions were published by Carl Gustav Jakob Jacobi and Niels Henrik Abel in 1827.

Related topics

• Möbius transformation
• Fuchsian group
• modular function
  • J-invariant
• modular form
• modular curve

External Links

• Symmetries of Period-Doubling Maps (http://www.linas.org/math/chap-takagi/chap-takagi.html) provides a relationship to fractals.