Modular group Gamma
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.
The modular group Γ is the group of linear fractional transformationss of the upper half of the complex plane which have the form
This group of transformations is isomorphic to the matrix 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
Definition
where a, b, c, and d are integers, and ad − bc = 1.
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.
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 Jacobi and Abel in 1827.
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
History
Relationship to hyperbolic geometry
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:
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.
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
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.
The modular group can be shown to be generated by the two transformations
There are other possible choices of generators - for example, using the identity the generator T can be replaced by the transformation TS i.e. S followed by T. So the modular group is also generated by S and
[...include here at least some mention of quadratic forms, the fundamental domain (modular curve) and modular forms...]
[...brief mention and definition of congruence subgroup, this really deserves its own article independent of Γ]Tessellation of the hyperbolic plane
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.Group-theoretic properties
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.
Noting that S has order 2 and U has order 3, this shows that the modular group is isomorphic to the free product of the cyclic groups C2 and C3.Applications to number theory
Congruence subgroups