Dessin D'enfant - Riemann Surfaces and Belyi Pairs

Riemann Surfaces and Belyi Pairs

The complex numbers, together with a special point designated as ∞, form a topological space known as the Riemann sphere. Any polynomial, and more generally any rational function p(x)/q(x) where p and q are polynomials, transforms the Riemann sphere by mapping it to itself. Consider, for example, the rational function

At most points of the Riemann sphere, this transformation is a local homeomorphism: it maps a small disk centered at any point in a one-to-one way into another disk. However, at certain critical points, the mapping is more complicated, and maps a disk centered at the point in a k-to-one way onto its image. The number k is known as the degree of the critical point and the transformed image of a critical point is known as a critical value. The example given above, ƒ, has the following critical points and critical values (some points of the Riemann sphere that, while not themselves critical, map to one of the critical values, are also included; these are indicated by having degree one):

critical point x	critical value ƒ(x)	degree
0	∞	1
1	0	3
9	0	1
3 + 2√3 ≈ 6.464	1	2
3 − 2√3 ≈ −0.464	1	2
∞	∞	3

One may form a dessin d'enfant from ƒ by placing black points at the preimages of 0 (that is, at 1 and 9), white points at the preimages of 1 (that is, at 3 ± 2√3), and arcs at the preimages of the line segment . This line segment has four preimages, two along the line segment from 1 to 9 and two forming a simple closed curve that loops from 1 to itself, surrounding 0; the resulting dessin is shown in the figure.

In the other direction, from this dessin, described as a combinatorial object without specifying the locations of the critical points, one may form a compact Riemann surface, and a map from that surface to the Riemann sphere, equivalent to the map from which the dessin was originally constructed. To do so, place a point labeled ∞ within each region of the dessin (shown as the red points in the second figure), and triangulate each region by connecting this point to the black and white points forming the boundary of the region, connecting multiple times to the same black or white point if it appears multiple times on the boundary of the region. Each triangle in the triangulation has three vertices labeled 0 (for the black points), 1 (for the white points), or ∞. For each triangle, substitute a half-plane, either the upper half-plane for a triangle that has 0, 1, and ∞ in counterclockwise order or the lower half-plane for a triangle that has them in clockwise order, and for every adjacent pair of triangles glue the corresponding half-planes together along the portion of their boundaries indicated by the vertex labels. The resulting Riemann surface can be mapped to the Riemann sphere by using the identity map within each half-plane. Thus, the dessin d'enfant formed from ƒ is sufficient to describe ƒ itself up to biholomorphism.

The same construction applies more generally when X is any Riemann surface and ƒ is a Belyi function; that is, a holomorphic function ƒ from X to the Riemann sphere having only 0, 1, and ∞ as critical values. A pair (X, ƒ) of this type is known as a Belyi pair. From any Belyi pair (X, ƒ) one can form a dessin d'enfant, drawn on the surface X, that has its black points at the preimages ƒ−1(0) of 0, its white points at the preimages ƒ−1(1) of 1, and its edges placed along the preimages ƒ−1 of the line segment . Conversely, any dessin d'enfant on any surface X can be used to define gluing instructions for a collection of halfspaces that together form a Riemann surface homeomorphic to X; mapping each halfspace by the identity to the Riemann sphere produces a Belyi function ƒ on X, and therefore leads to a Belyi pair (X, ƒ). Any two Belyi pairs (X, ƒ) that lead to combinatorially equivalent dessins d'enfants are biholomorphic, and Belyi's theorem implies that, for any compact Riemann surface X defined over the algebraic numbers, there is a Belyi function ƒ and a dessin d'enfant that provides a combinatorial description of both X and ƒ.