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| {{Group theory sidebar |Discrete}}
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| In [[mathematics]], the '''modular group Γ''' is a fundamental object of study in [[number theory]], [[geometry]], [[abstract algebra|algebra]], and many other areas of advanced mathematics. The modular group can be represented as a [[group (mathematics)|group]] of geometric transformations or as a group of [[matrix (math)|matrices]].
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| ==Definition==
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| The '''modular group Γ''' is the group of [[Möbius transformation|linear fractional transformation]]s of the [[upper half-plane|upper half of the complex plane]] which have the form
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| :<math>z\mapsto\frac{az+b}{cz+d}</math>
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| where ''a'', ''b'', ''c'', and ''d'' are [[integer]]s, and ''ad'' − ''bc'' = 1. The group operation is [[function composition]].
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| This group of transformations is isomorphic to the [[projective linear group|projective special linear group]] PSL(2, '''Z'''), which is the quotient of the 2-dimensional [[special linear group]] SL(2, '''Z''') over the integers by its [[center (group theory)|center]] {''I'', −''I''}. In other words, PSL(2, '''Z''') consists of all [[matrix (mathematics)|matrices]]
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| :<math>\begin{pmatrix} a & b \\ c & d \end{pmatrix}</math>
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| 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 [[matrix multiplication|multiplication of matrices]].
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| 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.
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| Some mathematical relations require the consideration of the group S*L(2, '''Z''') of matrices with determinant plus or minus one. (SL(2, '''Z''') is a subgroup of this group.) Similarly, PS*L(2, '''Z''') is the quotient group S*L(2,'''Z''')/{''I'', −''I''}. A 2 × 2 matrix with unit determinant is a [[symplectic matrix]], and thus SL(2, '''Z''') = Sp(2, '''Z'''), the [[symplectic group]] of 2x2 matrices.
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| One can also use the notation GL(2, '''Z''') for S*L(2, '''Z'''), because the inverse of an integer matrix exists and has integer coefficients if and only if it has determinant equal to ±1 (if the determinant neither zero nor ±1, the inverse will exist but have at least one non-integer coefficient). Alternatively, one may use the explicit notation SL<sup>±</sup>(2, '''Z''').
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| ==Number-theoretic properties==
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| The unit [[determinant]] of
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| :<math>\begin{pmatrix} a & b \\ c & d \end{pmatrix}</math>
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| 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
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| :<math>\frac{ap+bq}{cp+dq}</math>
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| 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 | |
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| :<math>\begin{pmatrix} a & b \\ c & d \end{pmatrix}\in\operatorname{SL}(2,\mathbf{Z})</math>
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| such that
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| :<math>r = ap+bq \quad \mbox{ and } \quad s=cp+dq.</math>
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| Elements of the modular group provide a symmetry on the two-dimensional [[period lattice|lattice]]. Let <math>\omega_1</math> and <math>\omega_2</math> be two complex numbers whose ratio is not real. Then the set of points
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| :<math>\Lambda (\omega_1, \omega_2)=\{ m\omega_1 +n\omega_2 : m,n\in \mathbf{Z} \}</math>
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| is a lattice of parallelograms on the plane. A different pair of vectors <math>\alpha_1</math> and <math>\alpha_2</math> will generate exactly the same lattice if and only if
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| :<math>\begin{pmatrix}\alpha_1 \\ \alpha_2 \end{pmatrix} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} \omega_1 \\ \omega_2 \end{pmatrix}</math>
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| for some matrix in S*L(2, '''Z'''). It is for this reason that [[doubly periodic function]]s, such as [[elliptic functions]], possess a modular group symmetry.
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| 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]]). 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.
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| If <math>p_{n-1}/q_{n-1}</math> and <math>p_{n}/q_{n}</math> are two successive convergents of a [[continued fraction]], then the matrix
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| :<math>\begin{pmatrix} p_{n-1} & p_{n} \\ q_{n-1} & q_{n} \end{pmatrix}</math> | |
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| belongs to S*L(2, '''Z'''). 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]] of order min(''b'', ''d''). Important special cases of continued fraction convergents include the [[Fibonacci number]]s 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 ===
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| The modular group can be shown to be [[generating set of a group|generated]] by the two transformations
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| :<math>S: z\mapsto -1/z</math>
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| :<math>T: z\mapsto z+1</math>
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| 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.
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| The generators ''S'' and ''T'' obey the relations ''S''<sup>2</sup> = 1 and (''ST'')<sup>3</sup> = 1. It can be shown <ref>{{cite journal | first = Roger C. | last= Alperin | title = PSL<sub>2</sub>(Z) = Z<sub>2</sub> * Z<sub>3</sub> | journal = Amer. Math. Monthly | volume = 100 | pages = 385–386 | date = April 1993 }}</ref> that these are a complete set of relations, so the modular group has the [[presentation of a group|presentation]]:
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| :<math>\Gamma \cong \langle S, T \mid S^2=I, (ST)^3=I \rangle</math>
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| This presentation describes the modular group as the rotational [[triangle group]] (2,3,∞) (∞ as there is no relation on ''T''), and it thus maps onto all triangle groups (2,3,''n'') by adding the relation ''T<sup>n</sup>'' = 1, which occurs for instance in the [[congruence subgroup]] Γ(''n'').
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| Using the generators ''S'' and ''ST'' instead of ''S'' and ''T'', this shows that the modular group is isomorphic to the [[free product]] of the [[cyclic group]]s C<sub>2</sub> and C<sub>3</sub>:
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| :<math>\Gamma \cong C_2 * C_3</math>
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| ===Braid group===
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| [[File:Braid-modular-group-cover.svg|thumb|376px|The [[braid group]] ''B''<sub>3</sub> is the [[universal central extension]] of the modular group.]]
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| The braid group ''B''<sub>3</sub> is the universal central extension of the modular group, with these sitting as lattices inside the (topological) universal covering group <math>\overline{\mathrm{SL}_2(\mathbf{R})} \to \mathrm{PSL}_2(\mathbf{R})</math>. Further, the modular group has trivial center, and thus the modular group is isomorphic to the [[quotient group]] of ''B''<sub>3</sub> modulo its [[center (group theory)|center]]; equivalently, to the group of [[inner automorphism]]s of ''B''<sub>3</sub>.
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| The braid group ''B''<sub>3</sub> in turn is isomorphic to the [[knot group]] of the [[trefoil knot]].
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| ===Quotients===
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| The quotients by congruence subgroups are of significant interest.
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| 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 ''T<sup>n</sup>'' = (''z'' ↦ ''z''+''n''). (2,3,5) is the group of [[icosahedral symmetry]], and the [[(2,3,7) triangle group]] (and associated tiling) is the cover for all [[Hurwitz surface]]s.
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| ==Relationship to hyperbolic geometry==
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| {{See also|PSL2(R)}}
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| The modular group is important because it forms a [[group (mathematics)|subgroup]] of the group of [[isometry|isometries]] of the [[hyperbolic geometry|hyperbolic plane]]. If we consider the [[upper half-plane]] model '''H''' of hyperbolic plane geometry, then the group of all
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| [[Orientation (mathematics)|orientation-preserving]] isometries of '''H''' consists of all [[Möbius transformation]]s of the form
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| :<math>z\mapsto \frac{az + b}{cz + d}</math>
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| where ''a'', ''b'', ''c'', and ''d'' are [[real number]]s and ''ad'' − ''bc'' = 1. Put differently, the group PSL(2, '''R''') [[group action|acts]] on the upper half-plane '''H''' according to the following formula:
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| :<math>\begin{pmatrix} a & b \\ c & d \end{pmatrix}\cdot z \,= \,\frac{az + b}{cz + d}</math>
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| This (left-)action is [[Sharply multiply transitive|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'''.
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| ===Tessellation of the hyperbolic plane===
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| [[File:ModularGroup-FundamentalDomain-01.png|thumb|600px|right|A typical fundamental domain for the action of Γ on the upper half-plane.]]
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| The modular group Γ acts on '''H''' as a [[discrete group|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 (group theory)|orbit]] of ''z''. This also means that we can construct [[fundamental domain]]s, which (roughly) contain exactly one representative from the orbit of every ''z'' in '''H'''. (Care is needed on the boundary of the domain.)
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| There are many ways of constructing a fundamental domain, but a common choice is the region
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| :<math>R = \left\{ z \in \mathbf{H}: \left| z \right| > 1,\, \left| \,\mbox{Re}(z) \,\right| < \frac{1}{2} \right\}</math>
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| 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.
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| By transforming this region in turn by each of the elements of the modular group, a [[tessellation|regular tessellation]] 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 metric|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 [[:Image:J-inv-phase.jpeg|J-invariant]], which is invariant under the modular group, and attains every complex number once in each triangle of these regions.
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| 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 {{Harv|Klein|1878/79a}},<ref name="lebruyn">{{citation
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| | last = le Bruyn | first = Lieven
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| | title = Dedekind or Klein ?
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| | date = 22 April 2008
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| | url = http://www.neverendingbooks.org/index.php/dedekind-or-klein.html
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| }}</ref> where it is credited to [[Richard Dedekind]], in reference to {{Harv|Dedekind|1877}}.<ref name="lebruyn" /><ref>{{Cite journal
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| | issn = 0002-9890
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| | volume = 108
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| | issue = 1
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| | pages = 70–76
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| | last = Stillwell
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| | first = John
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| | title = Modular Miracles
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| | journal = The American Mathematical Monthly| date = January 2001
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| | jstor = 2695682
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| }}</ref>
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| [[File:Morphing of modular tiling to 2 3 7 triangle tiling.gif|thumb|Visualization of the map (2,3,∞) → (2,3,7) by morphing the associated tilings.<ref name="westendorp">[http://www.xs4all.nl/~westy31/Geometry/Geometry.html#Modular Platonic tilings of Riemann surfaces: The Modular Group], [http://www.xs4all.nl/~westy31/ Gerard Westendorp]</ref>]]
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| 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.
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| {{Order i-3 tiling table}}
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| ==Congruence subgroups==
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| {{Main|Congruence subgroup}}
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| Important [[subgroup]]s of the modular group Γ, called ''[[congruence subgroup]]s'', are given by imposing [[congruence relation]]s on the associated matrices.
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| There is a natural [[homomorphism]] SL(2, '''Z''') → SL(2, '''Z'''/n'''Z''') given by reducing the entries [[modulo operation|modulo]] ''N''. This induces a homomorphism on the modular group PSL(2, '''Z''') → PSL(2, '''Z'''/n'''Z'''). The [[kernel (algebra)|kernel]] of this homomorphism is called the '''[[principal congruence subgroup]] of level ''N''''', denoted Γ(''N''). We have the following [[short exact sequence]]:
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| :<math>1\to\Gamma(N)\to\Gamma\to\mbox{PSL}(2,\mathbf{Z}/n\mathbf{Z})\to 1</math>.
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| Being the kernel of a homomorphism Γ(''N'') is a [[normal subgroup]] of the modular group Γ. The group Γ(''N'') is given as the set of all modular transformations
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| :<math>z\mapsto\frac{az+b}{cz+d}</math>
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| for which ''a'' ≡ ''d'' ≡ ±1 (mod ''N'') and ''b'' ≡ ''c'' ≡ 0 (mod ''N''). | |
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| The principal congruence subgroup of level 2, Γ(2), is also called the '''modular group Λ'''. Since PSL(2, '''Z'''/2'''Z''') is isomorphic to [[symmetric group|''S''<sub>3</sub>]], Λ is a subgroup of [[index of a subgroup|index]] 6. The group Λ consists of all modular transformations for which ''a'' and ''d'' are odd and ''b'' and ''c'' are even.
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| Another important family of congruence subgroups are the [[modular group Gamma0|modular group Γ<sub>0</sub>(''N'')]] defined as the set of all modular transformations for which ''c'' ≡ 0 (mod ''N''), or equivalently, {{Citation needed|date=July 2013}} as the subgroup whose matrices become [[upper triangular matrix|upper triangular]] upon reduction modulo ''N''. Note that Γ(''N'') is a subgroup of Γ<sub>0</sub>(''N''). The [[modular curve]]s associated with these groups are an aspect of [[monstrous moonshine]] – for a [[prime number]] ''p'', the modular curve of the normalizer is [[genus (mathematics)|genus]] zero if and only if ''p'' divides the [[order (group theory)|order]] of the [[monster group]], or equivalently, if ''p'' is a [[supersingular prime (moonshine theory)|supersingular prime]].
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| ==Dyadic monoid==
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| One important subset of the modular group is the '''dyadic monoid''', which is the [[monoid]] of all strings of the form ''ST<sup>k</sup>ST<sup>m</sup>ST<sup>n</sup>'' ... for positive integers ''k'', ''m'', ''n'', ... . This monoid occurs naturally in the study of [[fractal curve]]s, and describes the self-similarity symmetries of the [[Cantor function]], [[Minkowski's question mark function]], and the [[Koch snowflake|Koch curve]], each being a special case of the general [[de Rham curve]]. 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]].
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| ==Maps of the torus==
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| The group GL(2, '''Z''') is the linear maps preserving the standard lattice '''Z'''<sup>2</sup>, and SL(2, '''Z''') is the orientation-preserving maps preserving this lattice; they thus descend to [[self-homeomorphism]]s 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 [[Homotopy#Isotopy|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.
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| ==Hecke groups==
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| The modular group can be generalized to the '''Hecke groups,''' named for [[Erich Hecke]], and defined as follows.<ref>''Combinatorial group theory, discrete groups, and number theory,'' by Gerhard Rosenberger, Benjamin Fine, Anthony M. Gaglione, Dennis Spellman [http://books.google.com/books?id=5Unmxs7yeHwC&pg=PA65 p. 65]</ref>
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| The Hecke group ''H''<sub>''q''</sub> is the discrete group generated by
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| :<math>z \mapsto -1/z</math>
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| :<math>z \mapsto z + \lambda_q,</math>
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| where <math>\lambda_q=2\cos(\pi/q). \,</math>
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| The modular group Γ is isomorphic to ''H''<sub>3</sub> and they share properties and applications – for example, just as one has the [[free product]] of [[cyclic group]]s
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| :<math>\Gamma \cong C_2 * C_3,</math>
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| more generally one has | |
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| :<math>H_q \cong C_2 * C_q,</math>
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| which corresponds to the [[triangle group]] (2,''q'',∞). There is similarly a notion of principal congruence subgroups associated to principal ideals in '''Z'''[λ]. For small values of ''q'', one has:
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| :<math>\lambda_3 = 1,</math>
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| :<math>\lambda_4 = \sqrt{2},</math>
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| :<math>\lambda_5 = \tfrac{1}{2}(1+\sqrt{5}),</math>
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| :<math>\lambda_6 = \sqrt{3}.</math>
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| ==History==
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| 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 function]]s 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==
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| {{colbegin|3}}
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| *[[Möbius transformation]]
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| *[[Fuchsian group]]
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| *[[Bianchi group]]
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| *[[Kleinian group]]
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| *[[Hyperbolic tiling]]s
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| *[[modular function]]
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| *[[J-invariant]]
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| *[[modular form]]
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| *[[modular curve]]
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| *[[classical modular curve]]
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| *[[Poincaré half-plane model]]
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| *[[Minkowski's question mark function]]
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| *[[Mapping class group]]
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| {{colend}}
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| ==References==
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| {{Reflist}}
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| {{Refbegin}}
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| *Tom M. Apostol, ''Modular Functions and Dirichlet Series in Number Theory, Second Edition'' (1990), Springer, New York ISBN 0-387-97127-0 See chapter 2.
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| *{{Citation
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| |last=Klein | first = Felix | authorlink = Felix Klein
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| |ref={{Harvid|Klein|1878/79a}}
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| |title = Über die Transformation der elliptischen Funktionen und die Auflösung der Gleichungen fünften Grades (On the transformation of elliptic functions and ...)
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| |pages = 13–75 (in Oeuvres, Tome 3)
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| |journal = Math. Annalen
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| |volume = 14
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| |year = 1878/79
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| |url=http://mathdoc.emath.fr/cgi-bin/oetoc?id=OE_KLEIN__3
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| }}
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| * {{Citation | title = Schreiben an Herrn Borchard uber die Theorie der elliptische Modulfunktionen | first = Richard | last = Dedekind | authorlink = Richard Dedekind |date=September 1877 | journal = Crelle's Journal | volume = 83 | pages = 265–292}}.
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| {{Refend}}
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| {{Use dmy dates|date=September 2010}}
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| {{DEFAULTSORT:Modular Group}}
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| [[Category:Group theory]]
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| [[Category:Analytic number theory]]
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| [[Category:Modular forms]]
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