Modular group

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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.

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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 adbc = 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 adbc = 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. Note that SL(2, Z) is a subgroup of this group. Similarly, PS*L(2,Z) is the quotient group S*L(2,Z)/{I, −I}. Note that a 2x2 matrix with unit determinant is a symplectic matrix, and thus SL(2,Z)=Sp(2,Z), the symplectic group of 2x2 matrices.

Number-theoretic properties

The unit determinant implies that the fractions a/b, a/c, c/d and b/d are all irreducible, that is have no common factors. More generally, if p/q is an irreducible fraction, then

(a p+b q)/(c p+d q)

is irreducible. Any pair of irreducible fractions can be connected in this way.

Elements of the modular group provide a symmetry on the two-dimensional lattice. Let <math>\omega_1<math> and <math>\omega_2<math> be two complex numbers whose ratio is not real. Then the set of points <math>\Lambda (\omega_1, \omega_2)=\{ m\omega_1 +n\omega_2 : m,n\in \mathbb{Z} \}<math> is a lattice of parallelograms on the plane. A different pair of vectors <math>\alpha_1<math> and <math>\alpha_2<math> will generate exactly the same lattice if and only if

<math>\begin{pmatrix}\alpha_1 \\ \alpha_2 \end{pmatrix} =

\begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} \omega_1 \\ \omega_2 \end{pmatrix} <math> for some matrix in <math>S^*L(2,\mathbb{Z})<math>. It is for this reason that doubly-periodic functions, such as elliptic functions, possess a modular group symmetry.

The action of the modular group on the rational numbers can most easily be understood by envisioning a square grid, with grid point (p,q) corresponding to the fraction p/q. An irreducible fraction is one that is visible from the origin; the action of the modular group on a fraction never takes a visible (irreducible) to a hidden (reducible) one, and v.v.

If <math>p_{n-1}/q_{n-1}<math> and <math>p_{n}/q_{n}<math> are two successive convergents of a continued fraction, then the matrix

<math>\begin{pmatrix} p_{n-1} & p_{n} \\ q_{n-1} & q_{n} \end{pmatrix}<math>

belongs to <math>S^*L(2,\mathbb{Z})<math>. In particular, if bc − ad = 1 for positive integers a,b,c and d with a < b and c < d then ab and cd will be neighbours in the Farey sequence of order min(b,d). Important special cases of continued fraction convergents include the Fibonacci numbers and solutions to Pell's equation. In both cases, the numbers can be arranged to form a semigroup subset of the modular group.

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 adbc = 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

Missing image
ModularGroup-FundamentalDomain-01.png
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. Note that each such triangle has one vertex either at infinity or on the real axis Im(z)=0. This tiling can be extended to the Poincaré disk, where every hyperbolic triangle has one vertex on the boundary of the disk. The tiling of the Poincaré disk can be clearly seen the image of the J-invarient.

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,ZN) given by readings the entries modulo N. This induces a homomorphism on the modular group PSL(2,Z) → PSL(2,ZN). 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 ad ≡ ±1 (mod N) and bc ≡ 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).

Higher order representations

The modular group also has pairs of representations in GL(N,R). The N=3 representation can be understood to describe the self-symmetry of the blancmange curve, and higher N 's as the symmetries of similar curves. In this series of representations is another two-dimensional representation that gives the self-similarity symmetries of the Cantor function, the Minkowski question mark function, and the Koch curve, each being a special case of the general de Rham curve.

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

References

  • Tom M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Second Edition (1990), Springer, New York ISBN 0-387-97127-0 See chapter 2.
  • Linas Vepstas, Symmetries of Period-Doubling Maps (http://www.linas.org/math/chap-takagi/chap-takagi.html), (2004) constructs higher-dimensional representations of the modular group, and establishes a relationship to fractals.
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