Klein quartic

thumb|350px|The Klein quartic with the two dual [[Klein graphs (14-gon edges marked with the same number are equal.) The Klein quartic is a quotient of the heptagonal tiling (compare the 3-regular graph in green) and its dual triangular tiling (compare the 7-regular graph in violet).]]

In hyperbolic geometry, the Klein quartic, named after Felix Klein, is a compact Riemann surface of genus with the highest possible order automorphism group for this genus, namely order orientation-preserving automorphisms, and automorphisms if orientation may be reversed. As such, the Klein quartic is the Hurwitz surface of lowest possible genus; see Hurwitz's automorphisms theorem. Its (orientation-preserving) automorphism group is isomorphic to , the second-smallest non-abelian simple group after the alternating group A5. The quartic was first described in .

Klein's quartic occurs in many branches of mathematics, in contexts including representation theory, homology theory, Fermat's Last Theorem, and the Stark–Heegner theorem on imaginary quadratic number fields of class number one; see for a survey of properties.

Originally, the "Klein quartic" referred specifically to the subset of the complex projective plane defined by an algebraic equation. This has a specific Riemannian metric (that makes it a minimal surface in ), under which its Gaussian curvature is not constant. But more commonly (as in this article) it is now thought of as any Riemann surface that is conformally equivalent to this algebraic curve, and especially the one that is a quotient of the hyperbolic plane by a certain cocompact group that acts freely on by isometries. This gives the Klein quartic a Riemannian metric of constant curvature that it inherits from . This set of conformally equivalent Riemannian surfaces is precisely the same as all compact Riemannian surfaces of genus 3 whose conformal automorphism group is isomorphic to the unique simple group of order 168. This group is also known as , and also as the isomorphic group . By covering space theory, the group mentioned above is isomorphic to the fundamental group of the compact surface of genus .

Closed and open forms

It is important to distinguish two different forms of the quartic. The closed quartic is what is generally meant in geometry; topologically it has genus 3 and is a compact space. The open or "punctured" quartic is of interest in number theory; topologically it is a genus 3 surface with 24 punctures, and geometrically these punctures are cusps. The open quartic may be obtained (topologically) from the closed quartic by puncturing at the 24 centers of the tiling by regular heptagons, as discussed below. The open and closed quartics have different metrics, though they are both hyperbolic and complete – geometrically, the cusps are "points at infinity", not holes, hence the open quartic is still complete.

As an algebraic curve

The Klein quartic can be viewed as a projective algebraic curve over the complex numbers , defined by the following quartic equation in homogeneous coordinates on :

:<math>x^3y + y^3z + z^3x = 0.</math>

The locus of this equation in is the original Riemannian surface that Klein described.

Quaternion algebra construction

The compact Klein quartic can be constructed as the quotient of the hyperbolic plane by the action of a suitable Fuchsian group which is the principal congruence subgroup associated with the ideal <math>I=\langle \eta-2\rangle</math> in the ring of algebraic integers of the field where . Note the identity

:<math>(2-\eta)^3= 7(\eta-1)^2,</math>

exhibiting as a prime factor of 7 in the ring of algebraic integers.

The group is a subgroup of the (2,3,7) hyperbolic triangle group. Namely, is a subgroup of the group of elements of unit norm in the quaternion algebra generated as an associative algebra by the generators and relations

:<math>i^2=j^2=\eta, \qquad ij=-ji.</math>

One chooses a suitable Hurwitz quaternion order <math>\mathcal Q_{\mathrm{Hur</math> in the quaternion algebra, is then the group of norm 1 elements in <math>1+I\mathcal Q_{\mathrm{Hur</math>. The least absolute value of a trace of a hyperbolic element in is <math>\eta^2+3\eta+2</math>, corresponding the value 3.936 for the systole of the Klein quartic, one of the highest in this genus.

Tiling

thumb|The tiling of the quartic by reflection domains is a quotient of the [[3-7 kisrhombille.]]

The Klein quartic admits tilings connected with the symmetry group (a "regular map" Eigenvalues of the Klein quartic have been calculated to varying degrees of accuracy. The first 15 distinct positive eigenvalues are shown in the following table, along with their multiplicities.

{| class="wikitable"

|+ Numerical computations of the first 15 positive eigenvalues of the Klein quartic

! Eigenvalue

! Numerical value

! Multiplicity

| <math>\lambda_0</math>

| 0

| 1

| <math>\lambda_1</math>

| 2.67793

| 8

| <math>\lambda_2</math>

| 6.62251

| 7

| <math>\lambda_3</math>

| 10.8691

| 6

| <math>\lambda_4</math>

| 12.1844

| 8

| <math>\lambda_5</math>

| 17.2486

| 7

| <math>\lambda_6</math>

| 21.9705

| 7

| <math>\lambda_7</math>

| 24.0811

| 8

| <math>\lambda_8</math>

| 25.9276

| 6

| <math>\lambda_9</math>

| 30.8039

| 6

| <math>\lambda_{10}</math>

| 36.4555

| 8

| <math>\lambda_{11}</math>

| 37.4246

| 8

| <math>\lambda_{12}</math>

| 41.5131

| 6

| <math>\lambda_{13}</math>

| 44.8884

| 8

| <math>\lambda_{14}</math>

| 49.0429

| 6

| <math>\lambda_{15}</math>

| 50.6283

| 6

3-dimensional models

thumb|An animation by [[Greg Egan showing an embedding of Klein's Quartic Curve in three dimensions, starting in a form that has the symmetries of a tetrahedron, and turning inside out to demonstrate a further symmetry.]]

The Klein quartic cannot be realized as a 3-dimensional figure, in the sense that no 3-dimensional figure has (rotational) symmetries equal to , since does not embed as a subgroup of (or ) – it does not have a (non-trivial) 3-dimensional linear representation over the real numbers.

However, many 3-dimensional models of the Klein quartic have been given, starting in Klein's original paper, which seek to demonstrate features of the quartic and preserve the symmetries topologically, though not all geometrically. The resulting models most often have either tetrahedral (order 12) or octahedral (order 24) symmetries; the remaining order 7 symmetry cannot be as easily visualized, and in fact is the title of Klein's paper.

thumb|upright|The Eightfold Way – sculpture by [[Helaman Ferguson and accompanying book.]]

Most often, the quartic is modeled either by a smooth genus 3 surface with tetrahedral symmetry (replacing the edges of a regular tetrahedron with tubes/handles yields such a shape), which have been dubbed "tetruses", and the models are more complex than the triangular ones because the complexity is reflected in the shapes of the (non-flexible) heptagonal faces, rather than in the (flexible) vertices. the snub cube, The small cubicuboctahedron immersion is obtained by joining some of the triangles (2 triangles form a square, 6 form an octagon), which can be visualized by coloring the triangles (the corresponding tiling is topologically but not geometrically the 3 4 4 tiling). This immersion can also be used to geometrically construct the Mathieu group M24 by adding to PSL(2,7) the permutation which interchanges opposite points of the bisecting lines of the squares and octagons. That is, the quotient map is ramified over the points , and ; dividing by 1728 yields a Belyi function (ramified at , and ), where the 56 vertices (black points in dessin) lie over 0, the midpoints of the 84 edges (white points in dessin) lie over 1, and the centers of the 24 heptagons lie over infinity. The resulting dessin is a "platonic" dessin, meaning edge-transitive and "clean" (each white point has valence 2).

The Klein quartic is related to various other Riemann surfaces.

Geometrically, it is the smallest Hurwitz surface (lowest genus); the next is the Macbeath surface (genus 7), and the following is the First Hurwitz triplet (3 surfaces of genus 14). More generally, it is the most symmetric surface of a given genus (being a Hurwitz surface); in this class, the Bolza surface is the most symmetric genus 2 surface, while Bring's surface is a highly symmetric genus 4 surface – see isometries of Riemann surfaces for further discussion.

Algebraically, the (affine) Klein quartic is the modular curve X(7) and the projective Klein quartic is its compactification, just as the dodecahedron (with a cusp in the center of each face) is the modular curve X(5); this explains the relevance for number theory.

More subtly, the (projective) Klein quartic is a Shimura curve (as are the Hurwitz surfaces of genus 7 and 14), and as such parametrizes principally polarized abelian varieties of dimension 6.

More exceptionally, the Klein quartic forms part of a "trinity" in the sense of Vladimir Arnold, which can also be described as a McKay correspondence. In this collection, the projective special linear groups PSL(2,5), PSL(2,7), and PSL(2,11) (orders 60, 168, 660) are analogous. Note that 4 × 5 × 6/2 = 60, 6 × 7 × 8/2 = 168, and 10 × 11 × 12/2 = 660. These correspond to icosahedral symmetry (genus 0), the symmetries of the Klein quartic (genus 3), and the buckyball surface (genus 70). These are further connected to many other exceptional phenomena, which is elaborated at "trinities".

Closed and open forms

As an algebraic curve

Quaternion algebra construction

Tiling

3-dimensional models

See also

References

Literature

External links

Closed and open forms

As an algebraic curve

Quaternion algebra construction

Tiling

3-dimensional models

Related Riemann surfaces

See also

References

Literature

External links