# Hurwitz's automorphisms theorem

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.[1] 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 SL2(p) of order p3−p.

Contents 1 Interpretation in terms of hyperbolicity 2 Statement and proof 3 The idea of another proof and construction of the Hurwitz surfaces 3.1 Construction 4 Examples of Hurwitz groups and surfaces 5 Automorphism groups in low genus 6 See also 7 Notes 8 References 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 {displaystyle X} be a smooth connected Riemann surface of genus {displaystyle ggeq 2} . Then its automorphism group {displaystyle mathrm {Aut} (X)} has size at most {displaystyle 84(g-1)} Proof: Assume for now that {displaystyle G=mathrm {Aut} (X)} is finite (we'll prove this at the end). Consider the quotient map {displaystyle Xto X/G} . Since {displaystyle G} acts by holomorphic functions, the quotient is locally of the form {displaystyle zto z^{n}} and the quotient {displaystyle X/G} is a smooth Riemann surface. The quotient map {displaystyle Xto X/G} is a branched cover, and we will see below that the ramification points correspond to the orbits that have a non trivial stabiliser. Let {displaystyle g_{0}} be the genus of {displaystyle X/G} . By the Riemann-Hurwitz formula, {displaystyle 2g-2 = |G|cdot left(2g_{0}-2+sum _{i=1}^{k}left(1-{frac {1}{e_{i}}}right)right)} where the sum is over the {displaystyle k} ramification points {displaystyle p_{i}in X/G} for the quotient map {displaystyle Xto X/G} . The ramification index {displaystyle e_{i}} at {displaystyle p_{i}} is just the order of the stabiliser group, since {displaystyle e_{i}f_{i}=deg(X/,X/G)} where {displaystyle f_{i}} the number of pre-images of {displaystyle p_{i}} (the number of points in the orbit), and {displaystyle deg(X/,X/G)=|G|} . By definition of ramification points, {displaystyle e_{i}geq 2} for all {displaystyle k} ramification indices. Now call the righthand side {displaystyle |G|R} and since {displaystyle ggeq 2} we must have {displaystyle R>0} . Rearranging the equation we find: If {displaystyle g_{0}geq 2} then {displaystyle Rgeq 2} , and {displaystyle |G|leq (g-1)} If {displaystyle g_{0}=1} , then {displaystyle kgeq 1} and {displaystyle Rgeq 0+1-1/2=1/2} so that {displaystyle |G|leq 4(g-1)} , If {displaystyle g_{0}=0} , then {displaystyle kgeq 3} and if {displaystyle kgeq 5} then {displaystyle Rgeq -2+k(1-1/2)geq 1/2} , so that {displaystyle |G|leq 4(g-1)} if {displaystyle k=4} then {displaystyle Rgeq -2+4-1/2-1/2-1/2-1/3=1/6} , so that {displaystyle |G|leq 12(g-1)} , if {displaystyle k=3} then write {displaystyle e_{1}=p,,e_{2}=q,,e_{3}=r} . We may assume {displaystyle 2leq pleq q leq r} . if {displaystyle pgeq 3} then {displaystyle Rgeq -2+3-1/3-1/3-1/4=1/12} so that {displaystyle |G|leq 24(g-1)} , if {displaystyle p=2} then if {displaystyle qgeq 4} then {displaystyle Rgeq -2+3-1/2-1/4-1/5=1/20} so that {displaystyle |G|leq 40(g-1)} , if {displaystyle q=3} then {displaystyle Rgeq -2+3-1/2-1/3-1/7=1/42} so that {displaystyle |G|leq 84(g-1)} .

In conclusion, {displaystyle |G|leq 84(g-1)} .

To show that {displaystyle G} is finite, note that {displaystyle G} acts on the cohomology {displaystyle H^{*}(X,mathbf {C} )} preserving the Hodge decomposition and the lattice {displaystyle H^{1}(X,mathbf {Z} )} .