Editing Hurwitz's automorphisms theorem (section)

== The idea of another 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 [[covering space|covered]] by the [[Hyperbolic space|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 [[tessellation|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 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).

=== Construction ===
[[File:3-7 kisrhombille.svg|thumb|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 tiling]]s 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.
{| class="wikitable"
|[[File:Heptagonal tiling.svg|100px]]<br>[[order-3 heptagonal tiling]]
|[[File:Order-7 triangular tiling.svg|100px]]<br>[[order-7 triangular tiling]]
|}
[[Wythoff construction]]s yields further [[uniform tiling]]s, yielding [[Order-3 heptagonal tiling#Wythoff constructions from heptagonal and triangular tilings|eight uniform tilings]], including the two regular ones given here. These all descend to Hurwitz surfaces, yielding tilings of the surfaces (triangulation, tiling by heptagons, etc.).

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

:<math>a^2 = b^3 = (ab)^7 = 1,</math>

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.