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.
Interpretation in terms of hyperbolicityEdit
One of the fundamental themes in differential geometry is a trichotomy between the Riemannian manifolds of positive, zero, and negative curvatureK. 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;
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 proofEdit
Theorem: Let be a smooth connected Riemann surface of genus . Then its automorphism group has size at most
Proof: Assume for now that is finite (we'll prove this at the end).
Consider the quotient map . Since acts by holomorphic functions, the quotient is locally of the form and the quotient is a smooth Riemann surface. The quotient map is a branched cover, and we will see below that the ramification points correspond to the orbits that have a non trivial stabiliser. Let be the genus of .
where the sum is over the ramification points for the quotient map . The ramification index at is just the order of the stabiliser group, since where the number of pre-images of (the number of points in the orbit), and . By definition of ramification points, for all ramification indices.
Now call the righthand side and since we must have . Rearranging the equation we find:
It remains to prove that has finite kernel. In fact, we will prove is injective. Assume acts as the identity on . If is finite, then by the Lefschetz fixed-point theorem,
This is a contradiction, and so is infinite. Since is a closed complex sub variety of positive dimension and is a smooth connected curve (i.e. ), we must have . Thus is the identity, and we conclude that is injective and is finite.
Corollary of the proof: A Riemann surface of genus has automorphisms if and only if is a branched cover with three ramification points, of indices 2,3 and 7.
The idea of another proof and construction of the Hurwitz surfacesEdit
By the uniformization theorem, any hyperbolic surface X – i.e., the Gaussian curvature of X is equal to negative one at every point – is covered by the hyperbolic plane. The conformal mappings of the surface correspond to orientation-preserving automorphisms of the hyperbolic plane. By the Gauss–Bonnet theorem, the area of the surface is
A(X) = − 2π χ(X) = 4π(g − 1).
In order to make the automorphism group G of X as large as possible, we want the area of its fundamental domainD for this action to be as small as possible. If the fundamental domain is a triangle with the vertex angles π/p, π/q and π/r, defining a tiling of the hyperbolic plane, then p, q, and r are integers greater than one, and the area is
A(D) = π(1 − 1/p − 1/q − 1/r).
Thus we are asking for integers which make the expression
1 − 1/p − 1/q − 1/r
strictly positive and as small as possible. This minimal value is 1/42, and
1 − 1/2 − 1/3 − 1/7 = 1/42
gives a unique (up to permutation) triple of such integers. This would indicate that the order |G| of the automorphism group is bounded by
A(X)/A(D) ≤ 168(g − 1).
However, a more delicate reasoning shows that this is an overestimate by the factor of two, because the group G can contain orientation-reversing transformations. For the orientation-preserving conformal automorphisms the bound is 84(g − 1).
Hurwitz groups and surfaces are constructed based on the tiling of the hyperbolic plane by the (2,3,7) Schwarz triangle.
To obtain an example of a Hurwitz group, let us start with a (2,3,7)-tiling of the hyperbolic plane. Its full symmetry group is the full (2,3,7) triangle group generated by the reflections across the sides of a single fundamental triangle with the angles π/2, π/3 and π/7. Since a reflection flips the triangle and changes the orientation, we can join the triangles in pairs and obtain an orientation-preserving tiling polygon.
A Hurwitz surface is obtained by 'closing up' a part of this infinite tiling of the hyperbolic plane to a compact Riemann surface of genus g. This will necessarily involve exactly 84(g − 1) double triangle tiles.
The following two regular tilings have the desired symmetry group; the rotational group corresponds to rotation about an edge, a vertex, and a face, while the full symmetry group would also include a reflection. The polygons in the tiling are not fundamental domains – the tiling by (2,3,7) triangles refines both of these and is not regular.
From the arguments above it can be inferred that a Hurwitz group G is characterized by the property that it is a finite quotient of the group with two generators a and b and three relations
thus G is a finite group generated by two elements of orders two and three, whose product is of order seven. More precisely, any Hurwitz surface, that is, a hyperbolic surface that realizes the maximum order of the automorphism group for the surfaces of a given genus, can be obtained by the construction given.
This is the last part of the theorem of Hurwitz.
The smallest Hurwitz group is the projective special linear group PSL(2,7), of order 168, and the corresponding curve is the Klein quartic curve. This group is also isomorphic to PSL(3,2).
Next is the Macbeath curve, with automorphism group PSL(2,8) 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 smallest alternating group that is a Hurwitz group is A15.
^Technically speaking, there is an equivalence of categories between the category of compact Riemann surfaces with the orientation-preserving conformal maps and the category of non-singular complex projective algebraic curves with the algebraic morphisms.
^(Richter) Note each face in the polyhedron consist of multiple faces in the tiling – two triangular faces constitute a square face and so forth, as per this explanatory image.
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