Modular group
The modular group acts on the upper-half of the complex plane by linear fractional transformations.
The unit determinant of implies that the fractions a/b, a/c, c/d, b/d are all irreducible, that is having no common factors (provided the denominators are non-zero, of course).
Any pair of irreducible fractions can be connected in this way; that is, for any pair p/q and r/s of irreducible fractions, there exist elements such that Elements of the modular group provide a symmetry on the two-dimensional lattice.
A different pair of vectors α1 and α2 will generate exactly the same lattice if and only if for some matrix in GL(2, Z).
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 vice versa.
If pn−1/qn−1 and pn/qn are two successive convergents of a continued fraction, then the matrix belongs to GL(2, Z).
In particular, if bc − ad = 1 for positive integers a, b, c, d with a < b and c < d then a/b and c/d will be neighbours in the Farey sequence of order max(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.
The modular group can be shown to be generated by the two transformations 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 with respect to the imaginary axis, while T represents a unit translation to the right.
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 where a, b, c, d are real numbers.
There are many ways of constructing a fundamental domain, but a common choice is the region bounded by the vertical lines Re(z) = 1/2 and Re(z) = −1/2, and the circle |z| = 1.
in the upper half-plane gives an elliptic curve, namely the quotient of
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 Im(z) = 0.
The tiling of the Poincaré disk is given in a natural way by the J-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 (conventionally colored black and white), by adding an orientation-reversing map; the colors then correspond to orientation of the domain.
Adding in (x, y) ↦ (−x, y) and taking the right half of the region R (where Re(z) ≥ 0) yields the usual tessellation.
The kernel of this homomorphism is called the principal congruence subgroup of level N, denoted Γ(N).
It is easy to show that the trace of a matrix representing an element of Γ(N) cannot be −1, 0, or 1, so these subgroups are torsion-free groups.
The principal congruence subgroup of level 2, Γ(2), is also called the modular group Λ.
The modular curves associated with these groups are an aspect of monstrous moonshine – for a prime number p, the modular curve of the normalizer is genus zero if and only if p divides the order of the monster group, or equivalently, if p is a supersingular prime.
This monoid occurs naturally in the study of fractal curves, and describes the self-similarity symmetries of the Cantor function, Minkowski's question mark function, and the Koch snowflake, each being a special case of the general de Rham curve.
The algebraic properties of a matrix as an element of GL(2, Z) correspond to the dynamics of the induced map of the torus.
The modular group and its subgroups were first studied in detail by Richard Dedekind and by Felix Klein as part of his Erlangen programme in the 1870s.
However, the closely related elliptic functions were studied by Joseph Louis Lagrange in 1785, and further results on elliptic functions were published by Carl Gustav Jakob Jacobi and Niels Henrik Abel in 1827.
