Modular group
In mathematics, the modular group is the projective special linear group of matrices with integer coefficients and determinant 1. The matrices and are identified. The modular group acts on the upper-half of the complex plane by fractional linear transformations, and the name "modular group" comes from the relation to moduli spaces and not from modular arithmetic. Definition. The modular group is the group of linear fractional transformations of the upper half of the complex plane, which have the form where , , , 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 over the integers 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 the usual multiplication of matrices. 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. ( is a subgroup of this group.) Similarly, is the quotient group . A matrix  with unit determinant is a symplectic matrix, and thus , the symplectic group of matrices. Finding elements. To find an explicit matrix formula_3 in , begin with two coprime integers formula_4, and solve the determinant equationformula_5(Notice the determinant equation forces formula_4 to be coprime since otherwise there would be a factor formula_7 such that formula_8, formula_9, henceformula_10would have no integer solutions.) For example, if formula_11 then the determinant equation readsformula_12then taking formula_13 and formula_14 gives formula_15, henceformula_16is a matrix. Then, using the projection, these matrices define elements in . Number-theoretic properties. The unit determinant of implies that the fractions , , , are all irreducible , that is having no common factors (provided the denominators are non-zero, of course). More generally, if is an irreducible fraction, then is also irreducible (again, provided the denominator be non-zero). 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 (see Euclid's orchard). 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. 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 (the rationals with infinity) 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 transformations so that every element in the modular group can be represented (in a non-unique way) 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 (infinity as there is no relation on ), 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 (topological) 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 (and associated tiling) is the cover for  all Hurwitz surfaces. Presenting as a matrix group. The group formula_27 can be generated by the two matrices since The projection formula_30 turns these matrices into generators of formula_31, 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 all orientation-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 (roughly) contain exactly one representative from the orbit of every in . (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 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 formula_35 in the upper half-plane gives an elliptic curve, namely the quotient of formula_36 by the lattice generated by 1 and formula_35. 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 (conventionally colored black and white), 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 (where ) 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 (from modular group to triangle group) can be visualized in terms of this tiling (yielding a tiling on the modular curve), as depicted in the video at right. Congruence subgroups. Important subgroups of the modular group , called "congruence subgroups