Hurwitz's automorphisms theorem

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In mathematics, Hurwitz's automorphisms theorem bounds the group of automorphisms, via conformal mappings, of a compact Riemann surface of genus g > 1, telling us that the order of the group of such automorphisms is bounded by

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 of the equivalence of categories between compact Riemann surfaces and complex projective curves, a Hurwitz surface can also be called a Hurwitz curve. The theorem is due to Adolf Hurwitz, who proved it in 1893.

The conformal mappings of the Hurwitz surface correspond to orientation-preserving isogenies of the hyperbolic plane. In order to make the automorphism group as large as possible, we want the area of a triangular fundamental region to be as small as possible, which means we want

π(1 − 1/p − 1/q − 1/r)

to be as small as possible, where p, q, and r are positive integers defining the vertex angles π/p, π/q and π/r of a fundamental region for a tiling of the hyperbolic plane. Asking for integers which make

1 − 1/p − 1/q − 1/r

positive and as small as possible is a Diophantine question; to which the answer is

1 − 1/2 − 1/3 − 1/7 = 1/42.

Since a reflection flips the triangle, we join two of them and obtaining the orientation-preserving tiling polygon.

A Hurwitz group is characterized by the property that it is a finite group with generators a and b and relations including

a² = b³ = (ab)⁷ = 1;

in other words it is a finite group generated by two elements of orders two and three, whose product is of order seven. Hurwitz's result was that we obtain a Hurwitz surface, with the automorphisms maximum achieved, if and only if the automorphism group is a Hurwitz group.

The smallest Hurwitz group is the special linear group L₂(7), of order 168, and the corresponding curve is the Klein quartic curve.

Next is the Macbeath curve, with automorphism group L₂(8) of order 504. Many more finite simple groups are Hurwitz groups; for instance all but 64 of the alternating groups are Hurwitz groups, the largest non-Hurwitz example being of degree 167.

The sporadic Hurwitz groups are the Janko groups J₁, J₂ and J₄, the Fischer groups Fi₂₂ and Fi'₂₄, the Rudvalis group, the Held group, the Thompson group, the Harada-Norton group,the third Conway group Co₃, the Lyons group and best of all, the Monster.