Modular group
In mathematics, the modular group is the projective special linear group of matrices with integer coefficients and determinant, such that the matrices and are identified. The modular group acts on the upper-half of the complex plane by linear fractional transformations. The name "modular group" comes from the relation to moduli spaces, and not from modular arithmetic.
Definition
The modular group is the group of fractional linear transformations of the complex upper half-plane, which have the formwhere are integers, and. The group operation is function composition.
This group of transformations is isomorphic to the projective special linear group, which is the quotient of the 2-dimensional special linear group by its center. In other words, consists of all matrices
where are integers,, and pairs of matrices and are considered to be identical. The group operation is usual matrix multiplication.
Some authors define the modular group to be, and still others define the modular group to be the larger group.
Some mathematical relations require the consideration of the group of matrices with determinant plus or minus one. Similarly, is the quotient group.
Since all matrices with determinant 1 are symplectic matrices, then, the symplectic group of matrices.
Finding elements
To find an explicit matrixin, begin with two coprime integers, and solve the determinant equation.
For example, if then the determinant equation reads
then taking and gives. Hence
is a matrix in. Then, using the projection, these matrices define elements in.
Number-theoretic properties
The unit determinant ofimplies that the fractions , , , are all irreducible, that is having no common factors. More generally, if is an irreducible fraction, then
is also irreducible. Any pair of irreducible fractions can be connected in this way; that is, for any pair and of irreducible fractions, there exist elements
such that
Elements of the modular group provide a symmetry on the two-dimensional lattice. Let and be two complex numbers whose ratio is not real. Then the set of points
is a lattice of parallelograms on the plane. A different pair of vectors and will generate exactly the same lattice if and only if
for some matrix in. 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 corresponding to the fraction . An irreducible fraction is one that is visible from the origin; the action of the modular group on a fraction never takes a visible to a hidden one, and vice versa.
Note that any member of the modular group maps the projectively extended real line one-to-one to itself, and furthermore bijectively maps the projectively extended rational line to itself, the irrationals to the irrationals, the transcendental numbers to the transcendental numbers, the non-real numbers to the non-real numbers, the upper half-plane to the upper half-plane, et cetera.
If and are two successive convergents of a continued fraction, then the matrix
belongs to. In particular, if for positive integers,,, with and then and will be neighbours in the Farey sequence of order. 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
Presentation
The modular group can be shown to be generated by the two transformationsso that every element in the modular group can be represented by the composition of powers of and. Geometrically, represents inversion in the unit circle followed by reflection with respect to the imaginary axis, while represents a unit translation to the right.
The generators and obey the relations and. It can be shown that these are a complete set of relations, so the modular group has the presentation:
This presentation describes the modular group as the rotational triangle group , and it thus maps onto all triangle groups by adding the relation, which occurs for instance in the congruence subgroup.
Using the generators and instead of and, this shows that the modular group is isomorphic to the free product of the cyclic groups and :
Braid group
The braid group is the universal central extension of the modular group, with these sitting as lattices inside the universal covering group. Further, the modular group has a trivial center, and thus the modular group is isomorphic to the quotient group of modulo its center; equivalently, to the group of inner automorphisms of.The braid group in turn is isomorphic to the knot group of the trefoil knot.
Quotients
The quotients by congruence subgroups are of significant interest.Other important quotients are the triangle groups, which correspond geometrically to descending to a cylinder, quotienting the coordinate modulo, as. is the group of icosahedral symmetry, and the triangle group is the cover for all Hurwitz surfaces.
Presenting as a matrix group
The group can be generated by the two matricessince
The projection turns these matrices into generators of, with relations similar to the group presentation.
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 of hyperbolic plane geometry, then the group of allorientation-preserving isometries of consists of all Möbius transformations of the form
where,,, are real numbers. In terms of projective coordinates, the group acts on the upper half-plane by projectivity:
This action is faithful. Since is a subgroup of, the modular group is a subgroup of the group of orientation-preserving isometries of.
Tessellation of the hyperbolic plane
The modular group acts on as a discrete subgroup of, that is, for each in we can find a neighbourhood of which does not contain any other element of the orbit of. This also means that we can construct fundamental domains, which contain exactly one representative from the orbit of every in.There are many ways of constructing a fundamental domain, but a common choice is the region
bounded by the vertical lines and, and the circle. This region is a hyperbolic triangle. It has vertices at and, where the angle between its sides is, and a third vertex at infinity, where the angle between its sides is 0.
There is a strong connection between the modular group and elliptic curves. Each point in the upper half-plane gives an elliptic curve, namely the quotient of by the lattice generated by 1 and.
Two points in the upper half-plane give isomorphic elliptic curves if and only if they are related by a transformation in the modular group. Thus, the quotient of the upper half-plane by the action of the modular group is the so-called moduli space of elliptic curves: a space whose points describe isomorphism classes of elliptic curves. This is often visualized as the fundamental domain described above, with some points on its boundary identified.
The modular group and its subgroups are also a source of interesting tilings of the hyperbolic plane. By transforming this fundamental domain in turn by each of the elements of the modular group, a regular tessellation of the hyperbolic plane by congruent hyperbolic triangles known as the V6.6.∞ Infinite-order triangular tiling is created. Note that each such triangle has one vertex either at infinity or on the real axis.
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 is given in a natural way by the -invariant, which is invariant under the modular group, and attains every complex number once in each triangle of these regions.
This tessellation can be refined slightly, dividing each region into two halves, by adding an orientation-reversing map; the colors then correspond to orientation of the domain. Adding in and taking the right half of the region yields the usual tessellation. This tessellation first appears in print in, where it is credited to Richard Dedekind, in reference to.
The map of groups can be visualized in terms of this tiling, as depicted in the video at right.
Congruence subgroups
Important subgroups of the modular group, called congruence subgroups, are given by imposing congruence relations on the associated matrices.There is a natural homomorphism given by reducing the entries modulo. This induces a homomorphism on the modular group. The kernel of this homomorphism is called the principal congruence subgroup of level , denoted. We have the following short exact sequence:
Being the kernel of a homomorphism is a normal subgroup of the modular group. The group is given as the set of all modular transformations
for which and.
It is easy to show that the trace of a matrix representing an element of cannot be −1, 0, or 1, so these subgroups are torsion-free groups.
The principal congruence subgroup of level 2,, is also called the modular group . Since is isomorphic to, is a subgroup of index 6. The group consists of all modular transformations for which and are odd and and are even.
Another important family of congruence subgroups are the modular group defined as the set of all modular transformations for which, or equivalently, as the subgroup whose matrices become upper triangular upon reduction modulo. Note that is a subgroup of. The modular curves associated with these groups are an aspect of monstrous moonshine – for a prime number, the modular curve of the normalizer is genus zero if and only if divides the order of the monster group, or equivalently, if is a supersingular prime.