Hurwitz's Automorphisms Theorem - The Idea of A Proof and Construction of The Hurwitz Surfaces

The Idea of A Proof and Construction of The Hurwitz Surfaces

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 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

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. A remarkable fact is that 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).