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.

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. 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: 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.

be a smooth connected Riemann surface of genus

is a closed complex sub variety of positive dimension and

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 In order to make the automorphism group G of X as large as possible, we want the area of its fundamental domain D 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 Thus we are asking for integers which make the expression strictly positive and as small as possible.

This would indicate that the order |G| of the automorphism group is bounded by 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.

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.

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.

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.

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.

Next is the Macbeath curve, with automorphism group PSL(2,8) of order 504.

For lower ranks, fewer such groups are Hurwitz.

For np the order of p modulo 7, one has that PSL(2,q) is Hurwitz if and only if either q=7 or q = pnp.

Similarly, many groups of Lie type are Hurwitz.

The finite classical groups of large rank are Hurwitz, (Lucchini & Tamburini 1999).

Other families of exceptional and twisted Lie groups of low rank are shown to be Hurwitz in (Malle 1995).

In this range, there only exists a Hurwitz curve in genus g = 3 and g = 7 (sequence A179982 in the OEIS).

The concept of a Hurwitz surface can be generalized in several ways to a definition that has examples in all but a few genera.

This is possible for all orientable compact genera (see above section "Automorphism groups in low genus").