PSL(2,7) - Symmetries of The Klein Quartic

Symmetries of The Klein Quartic

The Klein quartic is the projective variety over the complex numbers C defined by the quartic polyomial

x3y + y3z + z3x = 0.

It is a compact Riemann surface of genus g = 3, and is the only one of these whose conformal automorphism group attains the Hurwitz bound of 84(g-1) for any genus g > 1. (This is the Hurwitz automorphisms theorem.) Such "Hurwitz surfaces" are rare; the next genus for which any exist is g = 7, and the next after that is g = 14.

As with all Hurwitz surfaces, the Klein quartic can be given a metric of constant negative curvature and then tiled with regular (hyperbolic) heptagons, as a quotient of the order-3 heptagonal tiling, with the symmetries of the surface as a Riemannian surface or algebraic curve exactly the same as the symmetries of the tiling. For the Klein quartic this yields a tiling by 24 heptagons, and the order of G is thus related to the fact that 24 × 7 = 168. Dually, it can be tiled with 56 equilateral triangles, with 24 vertices, each of degree 7, as a quotient of the order-7 triangular tiling.

Klein's quartic arises in many fields of mathematics, including representation theory, homology theory, octonion multiplication, Fermat's last theorem, and Stark's theorem on imaginary quadratic number fields of class number 1.