Hurwitz's automorphisms theorem

The set of Hurwitz automorphism on (after Adolf Hurwitz, 1893) is a statement of function theory. He says that the automorphism group of a compact hyperbolic Riemann surface is finite, and is a dependent only on topological properties of upper bound on their size.

Statement

( Ie are homeomorphic to a sphere, the "Henkel " glued ) Let be a compact Riemann surface of genus. Then the group of holomorphic automorphisms is finite and contains at most elements.

For the cases ( the Riemann sphere with infinite automorphism ) and ( torus, also with infinite automorphism group ) assessment does not apply. The validity of the estimate for is related to the fact that the universal covering of these areas is the hyperbolic half-plane, which is no longer true for.

Example

The Klein quartic, defined by the equation, regarded as a subset of the projective space is a Riemann surface of genus. Your automorphism group is isomorphic to and consists of elements.