Modular group

 ( Group Theory ) 

In mathematics, the modular group is the projective special linear group $\backslash operatorname\{PSL\}(2,\backslash mathbb\; Z)$ of $2\backslash times\; 2$ matrices with integer coefficients and determinant $1$, such that the matrices $A$ and $-A$ 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 , and not from modular arithmetic.

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

$z\backslash mapsto\backslash frac\{az+b\}\{cz+d\},$

where $a,b,c,d$ are integers, and $ad-bc=1$. The group operation is function composition.

This group of transformations is isomorphic to the projective special linear group $\backslash operatorname\{PSL\}(2,\backslash mathbb\; Z)$, which is the quotient of the 2-dimensional special linear group $\backslash operatorname\{SL\}(2,\backslash mathbb\; Z)$ by its center $\backslash \{I,-I\backslash \}$. In other words, $\backslash operatorname\{PSL\}(2,\backslash mathbb\; Z)$ consists of all matrices

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

Some authors define the modular group to be $\backslash operatorname\{PSL\}(2,\backslash mathbb\; Z)$, and still others define the modular group to be the larger group $\backslash operatorname\{SL\}(2,\backslash mathbb\; Z)$.

Some mathematical relations require the consideration of the group $\backslash operatorname\{GL\}(2,\backslash mathbb\; Z)$ of matrices with determinant plus or minus one. ($\backslash operatorname\{SL\}(2,\backslash mathbb\; Z)$ is a subgroup of this group.) Similarly, $\backslash operatorname\{PGL\}(2,\backslash mathbb\; Z)$ is the quotient group $\backslash operatorname\{GL\}(2,\backslash mathbb\; Z)/\backslash \{I,-I\backslash \}$.

Since all $2\backslash times\; 2$ matrices with determinant 1 are symplectic matrices, then $\backslash operatorname\{SL\}(2,\backslash mathbb\; Z)=\backslash operatorname\{Sp\}(2,\backslash Z)$, the symplectic group of $2\backslash times\; 2$ matrices.

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Finding elements To find an explicit matrix

in $\backslash operatorname\{SL\}(2,\backslash mathbb\; Z)$, begin with two coprime integers $a,b$, and solve the determinant equation $ay-bx\; =\; 1$.

For example, if $a\; =\; 7,\; \backslash text\{\; \}\; b\; =6$ then the determinant equation reads

$7y-6x\; =\; 1,$

then taking $y\; =\; -5$ and $x\; =\; -6$ gives $-35\; -\; (-36)\; =\; 1$. Hence

$\backslash begin\{pmatrix\}$

7 & -6 \\ 6 & -5 \end{pmatrix} is a matrix in $\backslash operatorname\{SL\}(2,\; \backslash mathbb\; Z)$. Then, using the projection, these matrices define elements in $\backslash operatorname\{PSL\}(2,\backslash mathbb\; Z)$.

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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

$\backslash frac\{ap+bq\}\{cp+dq\}$

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

$r\; =\; ap+bq\; \backslash quad\; \backslash mbox\{\; and\; \}\; \backslash quad\; s=cp+dq.$

Elements of the modular group provide a symmetry on the two-dimensional period lattice. Let and be two whose ratio is not real. Then the set of points

$\backslash Lambda\; (\backslash omega\_1,\; \backslash omega\_2)=\backslash \{\; m\backslash omega\_1\; +n\backslash omega\_2\; :\; m,n\backslash in\; \backslash mathbb\; Z\; \backslash \}$

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 and solutions to Pell's equation. In both cases, the numbers can be arranged to form a semigroup subset of the modular group.

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Group-theoretic properties

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Presentation The modular group can be shown to be generated by the two transformations

$\backslash begin\{align\}$

S &: z\mapsto -\frac1z \\ T &: z\mapsto z+1 \end{align}

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:

$\backslash Gamma\; \backslash cong\; \backslash left\backslash langle\; S,\; T\; \backslash mid\; S^2=I,\; \backslash left(ST\backslash right)^3=I\; \backslash right\backslash rangle$

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 and :

$\backslash Gamma\; \backslash cong\; C\_2\; *\; C\_3$

File:Sideway.gif|The action of on File:Turnovergif.gif|The action of on

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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.

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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 .

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Presenting as a matrix group The group $\backslash text\{SL\}\_2(\backslash mathbb\{Z\})$ can be generated by the two matrices

$S\; =\; \backslash begin\{pmatrix\}$

0 & -1 \\ 1 & 0 \end{pmatrix}, \text{ } T = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}

since

$S^2\; =\; -I\_2,\; \backslash text\{\; \}$

(ST)^3 = \begin{pmatrix} 0 & -1 \\ 1 & 1 \end{pmatrix}^3 = -I_2

The projection $\backslash text\{SL\}\_2(\backslash mathbb\{Z\})\; \backslash to\; \backslash text\{PSL\}\_2(\backslash mathbb\{Z\})$ turns these matrices into generators of $\backslash text\{PSL\}\_2(\backslash mathbb\{Z\})$, with relations similar to the group presentation.

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Relationship to hyperbolic geometry The modular group is important because it forms a subgroup of the group of isometry 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

$z\backslash mapsto\; \backslash frac\{az\; +\; b\}\{cz\; +\; d\}$

where , , , are . 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 .

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Tessellation of the hyperbolic plane The modular group acts on $\backslash mathbb\; H$ as a discrete group of $\backslash operatorname\{PSL\}(2,\backslash mathbb\; R)$, that is, for each in $\backslash mathbb\; H$ 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 $z$ in the upper half-plane gives an elliptic curve, namely the quotient of $\backslash mathbb\{C\}$ by the lattice generated by 1 and $z$. 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 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 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 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.

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Congruence subgroups Important 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 operation . 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:

$1\backslash to\backslash Gamma(N)\backslash to\backslash Gamma\backslash to\backslash operatorname\{PSL\}(2,\; \backslash mathbb\; Z/N\backslash mathbb\; Z)\; \backslash to\; 1.$

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

$z\backslash mapsto\backslash frac\{az+b\}\{cz+d\}$

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. (There are other torsion-free subgroups.)

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 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.

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Dyadic monoid One important subset of the modular group is the dyadic monoid, which is the monoid of all strings of the form for positive integers . This monoid occurs naturally in the study of , 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 monoid also has higher-dimensional linear representations; for example, the representation can be understood to describe the self-symmetry of the blancmange curve.

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Maps of the torus The group is the linear maps preserving the standard lattice , and is the orientation-preserving maps preserving this lattice; they thus descend to self-homeomorphisms of the torus (SL mapping to orientation-preserving maps), and in fact map isomorphically to the (extended) mapping class group 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 correspond to the dynamics of the induced map of the torus.

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Hecke groups The modular group can be generalized to the Hecke groups, named for Erich Hecke, and defined as follows.

(2026). 9780821839850, American Mathematical Society. . ISBN 9780821839850

The Hecke group with , is the discrete group generated by

$\backslash begin\{align\}$

z &\mapsto -\frac1z \\ z &\mapsto z + \lambda_q, \end{align}

where . For small values of , one has:

$\backslash begin\{align\}$

\lambda_3 &= 1, \\ \lambda_4 &= \sqrt{2}, \\ \lambda_5 &= \frac{1+\sqrt{5}}{2}, \\ \lambda_6 &= \sqrt{3}, \\ \lambda_8 &= \sqrt{2+\sqrt{2}}. \end{align}

The modular group is isomorphic to and they share properties and applications – for example, just as one has the free product of

$\backslash Gamma\; \backslash cong\; C\_2\; *\; C\_3,$

more generally one has

$H\_q\; \backslash cong\; C\_2\; *\; C\_q,$

which corresponds to the triangle group . There is similarly a notion of principal congruence subgroups associated to principal ideals in .

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History 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.

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See also

• Bianchi group
• Classical modular curve
• Fuchsian group
• J-invariant
• Kleinian group
• Mapping class group
• Minkowski's question-mark function
• Möbius transformation
• Modular curve
• Modular form
• Kuṭṭaka
• Poincaré half-plane model
• Uniform tilings in hyperbolic plane

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Notes

• Tom M. Apostol (1990). 9780387971278, Springer. ISBN 9780387971278
• .

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