Hurwitz's theorem on automorphism groups

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

statement

Let be a compact Riemann surface of gender (i.e. homeomorphic to a sphere with "handles" attached). Then the group of holomorphic automorphisms is finite and contains at most elements. ${\displaystyle X}$ ${\displaystyle g\geq 2}$ ${\displaystyle S^{2}}$ ${\displaystyle g\geq 2}$ ${\displaystyle \mathrm {Aut} (X)=\left\{f\colon X\to X\;{\mbox{bijektiv, holomorph}}\right\}}$ ${\displaystyle 84\cdot (g-1)}$

For the cases (the Riemann number sphere with infinite automorphism group) and ( torus , also with infinite automorphism group) the estimate does not apply. The validity of the estimate for is related to the fact that the universal superposition of these surfaces is the hyperbolic half-plane , which is no longer true for. ${\displaystyle g=0}$ ${\displaystyle \mathbb {P} ^{1}\mathbb {C} }$ ${\displaystyle g=1}$ ${\displaystyle g\geq 2}$ ${\displaystyle \mathbb {H} }$ ${\displaystyle g=0,1}$

example

The Klein quartic , defined by the equation , as a subset of projective space considered is a Riemannian surface of genus . Your automorphism group is isomorphic to and consists of elements. ${\displaystyle x^{3}y+y^{3}z+z^{3}x=0}$ ${\displaystyle \mathbb {P} ^{2}\mathbb {C} }$ ${\displaystyle 3}$ ${\displaystyle PSL(2,7)}$ ${\displaystyle 168=84\cdot (3-1)}$

literature

• A. Hurwitz: About algebraic structures with clear transformations in themselves. In: Math. Ann. Volume 41, 1893, pp. 403-442.