Concept
In hyperbolic geometry, the Klein quartic, named after Felix Klein, is a compact Riemann surface of genus 3 with the highest possible order automorphism group for this genus, namely order 168 orientation-preserving automorphisms, and 168 × 2 = 336 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 PSL(2, 7), 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, octonion multiplication, 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 P2(C) defined by an algebraic equation. This has a specific Riemannian metric (that makes it a minimal surface in P2(C)), 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 H2 by a certain cocompact group G that acts freely on H2 by isometries. This gives the Klein quartic a Riemannian metric of constant curvature −1 that it inherits from H2. 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 PSL(2, 7), and also as the isomorphic group PSL(3, 2). By covering space theory, the group G mentioned above is isomorphic to the fundamental group of the compact surface of genus 3. It is important to distinguish two different forms of the quartic.
