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In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the modular group Γ is a fundamental object of study in number theory

Number theory

Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

, geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, algebra

Abstract algebra

Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...

, and many other areas of advanced mathematics. The modular group can be represented as a group

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

of geometric transformations or as a group of matrices.

Definition

The modular group Γ is the group of linear fractional transformations of the upper half of the complex plane which have the form

where a, b, c, and d are integer

Integer

The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

s, and ad − bc = 1. The group operation is function composition

Function composition

In mathematics, function composition is the application of one function to the results of another. For instance, the functions and can be composed by computing the output of g when it has an argument of f instead of x...

.

This group of transformations is isomorphic to the projective special linear group

Projective linear group

In mathematics, especially in the group theoretic area of algebra, the projective linear group is the induced action of the general linear group of a vector space V on the associated projective space P...

PSL(2,Z), which is the quotient of the 2-dimensional special linear group

Special linear group

In mathematics, the special linear group of degree n over a field F is the set of n×n matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion....

over the integers by its center

Center (group theory)

In abstract algebra, the center of a group G, denoted Z,The notation Z is from German Zentrum, meaning "center". is the set of elements that commute with every element of G. In set-builder notation,...

{I, −I}. In other words, PSL(2,Z) consists of all matrices

Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

where a, b, c, and d are integers, ad − bc = 1, and pairs of matrices A and −A are considered to be identical. The group operation is the usual multiplication of matrices

Matrix multiplication

In mathematics, matrix multiplication is a binary operation that takes a pair of matrices, and produces another matrix. If A is an n-by-m matrix and B is an m-by-p matrix, the result AB of their multiplication is an n-by-p matrix defined only if the number of columns m of the left matrix A is the...

.

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

Symplectic group

In mathematics, the name symplectic group can refer to two different, but closely related, types of mathematical groups, denoted Sp and Sp. The latter is sometimes called the compact symplectic group to distinguish it from the former. Many authors prefer slightly different notations, usually...

of 2x2 matrices.

One can also use the notation GL(2,Z) for S^*L(2,Z), because an invertible integer matrix must have determinant +/-1; alternatively, one may use the explicit notation SL^±(2,Z).

Number-theoretic properties

The unit determinant

Determinant

In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...

implies that the fractions a/b, a/c, c/d and b/d are all irreducible, that is have no common factors (provided the denominators are non-zero, of course). More generally, if p/q is an irreducible fraction, then
/(c p+d q)

is also irreducible (again, provided the denominator be non-zero). Any pair of irreducible fractions can be connected in this way, i.e.: 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. 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 (p,q) corresponding to the fraction p/q (see Euclid's orchard

Euclid's orchard

In mathematics Euclid's orchard is an array of one-dimensional trees of unit height planted at the lattice points in one quadrant of a square lattice...

). 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

and

are two successive convergents of a continued fraction

Continued fraction

In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on...

, then the matrix

belongs to

. In particular, if bc − ad = 1 for positive integers a,b,c and d with a < b and c < d then ^a⁄_b and ^c⁄_d will be neighbours in the Farey sequence

Farey sequence

In mathematics, the Farey sequence of order n is the sequence of completely reduced fractions between 0 and 1 which, when in lowest terms, have denominators less than or equal to n, arranged in order of increasing size....

of order min(b,d). Important special cases of continued fraction convergents include the Fibonacci number

Fibonacci number

In mathematics, the Fibonacci numbers are the numbers in the following integer sequence:0,\;1,\;1,\;2,\;3,\;5,\;8,\;13,\;21,\;34,\;55,\;89,\;144,\; \ldots\; ....

s and solutions to Pell's equation

Pell's equation

Pell's equation is any Diophantine equation of the formx^2-ny^2=1\,where n is a nonsquare integer. The word Diophantine means that integer values of x and y are sought. Trivially, x = 1 and y = 0 always solve this equation...

. In both cases, the numbers can be arranged to form a semigroup

Semigroup

In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation. A semigroup generalizes a monoid in that there might not exist an identity element...

subset of the modular group.

Presentation

The modular group can be shown to be generated

Generating set of a group

In abstract algebra, a generating set of a group is a subset that is not contained in any proper subgroup of the group. Equivalently, a generating set of a group is a subset such that every element of the group can be expressed as the combination of finitely many elements of the subset and their...

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 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. It can be shown that these are a complete set of relations, so the modular group has the presentation

Presentation of a group

In mathematics, one method of defining a group is by a presentation. One specifies a set S of generators so that every element of the group can be written as a product of powers of some of these generators, and a set R of relations among those generators...

This presentation describes the modular group as the rotational triangle group

Triangle group

In mathematics, a triangle group is a group that can be realized geometrically by sequences of reflections across the sides of a triangle. The triangle can be an ordinary Euclidean triangle, a triangle on the sphere, or a hyperbolic triangle...

(2,3,∞) (∞ as there is no relation on T), and it thus maps onto all triangle groups (2,3,n) by adding the relation Tⁿ = 1, which occurs for instance in the congruence subgroup

Congruence subgroup

In mathematics, a congruence subgroup of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries. A very simple example would be invertible 2x2 integer matrices of determinant 1, such that the off-diagonal entries are even.An importance class of congruence...

Γ(n).

Using the generators S and ST instead of S and T, this shows that the modular group is isomorphic to the free product

Free product

In mathematics, specifically group theory, the free product is an operation that takes two groups G and H and constructs a new group G ∗ H. The result contains both G and H as subgroups, is generated by the elements of these subgroups, and is the “most general” group having these properties...

of the cyclic group

Cyclic group

In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...

s C₂ and C₃:

Braid group

The braid group B₃ 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 trivial center, and thus the modular group is isomorphic to the quotient group

Quotient group

In mathematics, specifically group theory, a quotient group is a group obtained by identifying together elements of a larger group using an equivalence relation...

of B₃ modulo its center

Center (group theory)

; equivalently, to the group of inner automorphism

Inner automorphism

In abstract algebra an inner automorphism is a functionwhich, informally, involves a certain operation being applied, then another one performed, and then the initial operation being reversed...

s of B₃.

The braid group B₃ in turn is isomorphic to the knot group

Knot group

In mathematics, a knot is an embedding of a circle into 3-dimensional Euclidean space. The knot group of a knot K is defined as the fundamental group of the knot complement of K in R3,\pi_1....

of the trefoil knot

Trefoil knot

In topology, a branch of mathematics, the trefoil knot is the simplest example of a nontrivial knot. The trefoil can be obtained by joining together the two loose ends of a common overhand knot, resulting in a knotted loop...

Quotients

The quotients by congruence subgroups are of significant interest.

Other important quotients are the (2,3,n) triangle groups, which correspond geometrically to descending to a cylinder, quotienting the x coordinate mod n, as Tⁿ = (z ↦ z+n). (2,3,5) is the group of icosahedral symmetry

Icosahedral symmetry

A regular icosahedron has 60 rotational symmetries, and a symmetry order of 120 including transformations that combine a reflection and a rotation...

, and the (2,3,7) triangle group

(2,3,7) triangle group

In the theory of Riemann surfaces and hyperbolic geometry, the triangle group is particularly important. This importance stems from its connection to Hurwitz surfaces, namely Riemann surfaces of genus g with the largest possible order, 84, of its automorphism group.A note on terminology – the "...

(and associated tiling) is the cover for all Hurwitz surface

Hurwitz surface

In Riemann surface theory and hyperbolic geometry, a Hurwitz surface, named after Adolf Hurwitz, is a compact Riemann surface with preciselyautomorphisms, where g is the genus of the surface. This number is maximal by virtue of Hurwitz's theorem on automorphisms...

Relationship to hyperbolic geometry

The modular group is important because it forms a subgroup

Group (mathematics)

of the group of isometries

Isometry

In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...

of the hyperbolic plane

Hyperbolic geometry

In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...

. If we consider the upper half-plane model H of hyperbolic plane geometry, then the group of all
orientation-preserving

Orientation (mathematics)

In mathematics, orientation is a notion that in two dimensions allows one to say when a cycle goes around clockwise or counterclockwise, and in three dimensions when a figure is left-handed or right-handed. In linear algebra, the notion of orientation makes sense in arbitrary dimensions...

isometries of H consists of all Möbius transformations of the form

where a, b, c, and d are real number

Real number

In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

s and ad − bc = 1. Put differently, the group PSL(2,R) acts

Group action

In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...

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.

Tessellation of the hyperbolic plane

The modular group Γ acts on H as a discrete subgroup

Discrete group

In mathematics, a discrete group is a group G equipped with the discrete topology. With this topology G becomes a topological group. A discrete subgroup of a topological group G is a subgroup H whose relative topology is the discrete one...

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 domain

Fundamental domain

In geometry, the fundamental domain of a symmetry group of an object is a part or pattern, as small or irredundant as possible, which determines the whole object based on the symmetry. More rigorously, given a topological space and a group acting on it, the images of a single point under the group...

s, 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

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 1/2 + i√3/2 and −1/2 + 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

Tessellation

A tessellation or tiling of the plane is a pattern of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of parts of the plane or of other surfaces. Generalizations to higher dimensions are also possible. Tessellations frequently appeared in the art...

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

Poincaré metric

In mathematics, the Poincaré metric, named after Henri Poincaré, is the metric tensor describing a two-dimensional surface of constant negative curvature. It is the natural metric commonly used in a variety of calculations in hyperbolic geometry or Riemann surfaces.There are three equivalent...

, 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 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 (Re(z) ≥ 0) yields the usual tessellation. This tessellation first appears in print in , where it is credited to Richard Dedekind

Richard Dedekind

Julius Wilhelm Richard Dedekind was a German mathematician who did important work in abstract algebra , algebraic number theory and the foundations of the real numbers.-Life:...

, in reference to .
The map of groups (2,3,∞) → (2,3,n) (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 subgroup

Subgroup

In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...

s of the modular group Γ, called congruence subgroup

Congruence subgroup

s, are given by imposing congruence relation

Congruence relation

In abstract algebra, a congruence relation is an equivalence relation on an algebraic structure that is compatible with the structure...

s on the associated matrices.

There is a natural homomorphism

Homomorphism

In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ὁμός meaning "same" and μορφή meaning "shape".- Definition :The definition of homomorphism depends on the type of algebraic structure under...

SL(2,Z) → SL(2,Z_N) given by reducing the entries modulo

Modulo operation

In computing, the modulo operation finds the remainder of division of one number by another.Given two positive numbers, and , a modulo n can be thought of as the remainder, on division of a by n...

N. This induces a homomorphism on the modular group PSL(2,Z) → PSL(2,Z_N). The kernel

Kernel (algebra)

In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. An important special case is the kernel of a matrix, also called the null space.The definition of kernel takes...

of this homomorphism is called the principal congruence subgroup of level N, denoted Γ(N). We have the following short exact sequence:

.
Being the kernel of a homomorphism Γ(N) is a normal subgroup

Normal subgroup

In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....

of the modular group Γ. The group Γ(N) is given as the set of all modular transformations

for which a ≡ d ≡ ±1 (mod N) and b ≡ c ≡ 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

Symmetric group

In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...

, Λ is a subgroup of index

Index of a subgroup

In mathematics, specifically group theory, the index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" of H that fill up G. For example, if H has index 2 in G, then intuitively "half" of the elements of G lie in H...

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 modular group Γ₀(N) defined as the set of all modular transformations for which c ≡ 0 (mod N), or equivalently, as the subgroup whose matrices become upper triangular upon reduction modulo N. Note that Γ(N) is a subgroup of Γ₀(N). The modular curves associated with these groups are an aspect of monstrous moonshine

Monstrous moonshine

In mathematics, monstrous moonshine, or moonshine theory, is a term devised by John Horton Conway and Simon P. Norton in 1979, used to describe the connection between the monster group M and modular functions .- History :Specifically, Conway and Norton, following an initial observationby John...

– for a prime p, the modular curve of the normalizer is genus zero if and only if p divides the order of the monster group

Monster group

In the mathematical field of group theory, the Monster group M or F1 is a group of finite order:...

, or equivalently, if p is a supersingular prime

Supersingular prime (moonshine theory)

In the mathematical branch of moonshine theory, a supersingular prime is a certain type of prime number.Namely, a supersingular prime is a prime divisor of the order of the Monster group M, the largest of the sporadic simple groups...

; see details at congruence subgroup.

Dyadic monoid

One important subset of the modular group is the dyadic monoid, which is the monoid

Monoid

In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they are naturally semigroups with identity. Monoids occur in several branches of mathematics; for...

of all strings of the form

for positive integers k,m,n,... . This monoid occurs naturally in the study of fractal curves, and describes the self-similarity symmetries of the Cantor function

Cantor function

In mathematics, the Cantor function, named after Georg Cantor, is an example of a function that is continuous, but not absolutely continuous. It is also referred to as the Devil's staircase.-Definition:See figure...

, Minkowski's question mark function

Minkowski's question mark function

In mathematics, the Minkowski question mark function, sometimes called the slippery devil's staircase and denoted by ?, is a function possessing various unusual fractal properties, defined by Hermann Minkowski in 1904...

, and the Koch curve

Koch snowflake

The Koch snowflake is a mathematical curve and one of the earliest fractal curves to have been described...

, each being a special case of the general de Rham curve

De Rham curve

In mathematics, a de Rham curve is a certain type of fractal curve named in honor of Georges de Rham.The Cantor function, Césaro curve, Minkowski's question mark function, the Lévy C curve, the blancmange curve and the Koch curve are all special cases of the general de Rham...

. The monoid also has higher-dimensional linear representations; for example, the N=3 representation can be understood to describe the self-symmetry of the blancmange curve

Blancmange curve

In mathematics, the blancmange curve is a fractal curve constructible by midpoint subdivision. It is also known as the Takagi curve, after Teiji Takagi who described it in 1903, or as the Takagi–Landsberg curve, a generalization of the curve. The name blancmange comes from its resemblance to a...

Maps of the torus

The group GL(2,Z) is the linear maps preserving the standard lattice Z², and SL(2,Z) is the orientation-preserving maps preserving this lattice; they thus descend to self-homeomorphisms of the torus

Torus

In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...

(SL mapping to orientation-preserving maps), and in fact map isomorphically to the (extended) mapping class group

Mapping class group

In mathematics, in the sub-field of geometric topology, the mapping class groupis an important algebraic invariant of a topological space. Briefly, the mapping class group is a discrete group of 'symmetries' of the space.-Motivation:...

of the torus, meaning that every self-homeomorphism of the torus is isotopic to a map of this form. The algebraic properties of a matrix as an element of GL(2,Z) correspond to the dynamics of the induced map of the torus.

Hecke groups

The modular group can be generalized to the Hecke groups, named for Erich Hecke

Erich Hecke

Erich Hecke was a German mathematician. He obtained his doctorate in Göttingen under the supervision of David Hilbert. Kurt Reidemeister and Heinrich Behnke were among his students....

, and defined as follows.

The Hecke group H_q is the discrete group generated by

and

where

The modular group is precisely

and they share properties and applications – for example, just as one has the free product

more generally one has

which corresponds to the triangle group

Triangle group

(2,q,∞). There is similarly a notion of principal congruence subgroups associated to principle ideals in

For small values of q, one has:

the golden ratio

Golden ratio

In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.61803398874989...

History

The modular group and its subgroups were first studied in detail by Richard Dedekind

Richard Dedekind

Julius Wilhelm Richard Dedekind was a German mathematician who did important work in abstract algebra , algebraic number theory and the foundations of the real numbers.-Life:...

and by Felix Klein

Felix Klein

Christian Felix Klein was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory...

as part of his Erlangen programme in the 1870s. However, the closely related elliptic function

Elliptic function

In complex analysis, an elliptic function is a function defined on the complex plane that is periodic in two directions and at the same time is meromorphic...

s were studied by Joseph Louis Lagrange

Joseph Louis Lagrange

Joseph-Louis Lagrange , born Giuseppe Lodovico Lagrangia, was a mathematician and astronomer, who was born in Turin, Piedmont, lived part of his life in Prussia and part in France, making significant contributions to all fields of analysis, to number theory, and to classical and celestial mechanics...

in 1785, and further results on elliptic functions were published by Carl Gustav Jakob Jacobi

Carl Gustav Jakob Jacobi

Carl Gustav Jacob Jacobi was a German mathematician, widely considered to be the most inspiring teacher of his time and is considered one of the greatest mathematicians of his generation.-Biography:...

and Niels Henrik Abel

Niels Henrik Abel

Niels Henrik Abel was a Norwegian mathematician who proved the impossibility of solving the quintic equation in radicals.-Early life:...

in 1827.