Hurwitz's automorphisms theorem

In mathematics, Hurwitz's automorphisms theorem bounds the order of the group of automorphisms, via orientation-preserving conformal mappings, of a compact Riemann surface of genus g > 1, stating that the number of such automorphisms cannot exceed 84(g − 1). A group for which the maximum is achieved is called a Hurwitz group, and the corresponding Riemann surface a Hurwitz surface. Because compact Riemann surfaces are synonymous with non-singular complex projective algebraic curves, a Hurwitz surface can also be called a Hurwitz curve. The theorem is named after Adolf Hurwitz, who proved it in (Hurwitz 1893). Hurwitz's bound also holds for algebraic curves over a field of characteristic 0, and over fields of positive characteristic p > 0 for groups whose order is coprime to p, but can fail over fields of positive characteristic p > 0 when p divides the group order. For example, the double cover of the projective line y2 = xp − x branched at all points defined over the prime field has genus g = (p − 1)/2 but is acted on by the group PGL2(p) of order p3 − p.

Interpretation in terms of hyperbolicity One of the fundamental themes in differential geometry is a trichotomy between the Riemannian manifolds of positive, zero, and negative curvature K. It manifests itself in many diverse situations and on several levels. In the context of compact Riemann surfaces X, via the Riemann uniformization theorem, this can be seen as a distinction between the surfaces of different topologies:

X a sphere, a compact Riemann surface of genus zero with K > 0; X a flat torus, or an elliptic curve, a Riemann surface of genus one with K = 0; and X a hyperbolic surface, which has genus greater than one and K < 0. While in the first two cases the surface X admits infinitely many conformal automorphisms (in fact, the conformal automorphism group is a complex Lie group of dimension three for a sphere and of dimension one for a torus), a hyperbolic Riemann surface only admits a discrete set of automorphisms. Hurwitz's theorem claims that in fact more is true: it provides a uniform bound on the order of the automorphism group as a function of the genus and characterizes those Riemann surfaces for which the bound is sharp.

Statement and proof Theorem: Let X {\displaystyle X} be a smooth compact Riemann surface of genus g ≥ 2 {\displaystyle g\geq 2} . Then its automorphism group Aut ⁡ ( X ) {\displaystyle \operatorname {Aut} (X)} has size at most 84 ( g − 1 ) {\displaystyle 84(g-1)} . Proof: Assume for now that G = Aut ⁡ ( X ) {\displaystyle G=\operatorname {Aut} (X)} is finite (this will be proved at the end).

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