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).
Read more about this topic: Hurwitz's Automorphisms Theorem
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