Dessin d'enfant

In mathematics, a dessin d'enfant is a type of graph embedding used to study Riemann surfaces and to provide combinatorial invariants for the action of the absolute Galois group of the rational numbers.

A dessin d'enfant is a graph, with its vertices colored alternately black and white, embedded in an oriented surface that, in many cases, is simply a plane.

The surface and the embedding may be described combinatorially using a rotation system, a cyclic order of the edges surrounding each vertex of the graph that describes the order in which the edges would be crossed by a path that travels clockwise on the surface in a small loop around the vertex.

Recognizable modern dessins d'enfants and Belyi functions were used by Felix Klein.

The three surfaces constructed in this way from these three groups were much later shown to be closely related through the phenomenon of trinity.

Dessins d'enfant in their modern form were then rediscovered over a century later and named by Alexander Grothendieck in 1984 in his Esquisse d'un Programme.

[5] Zapponi (2003) quotes Grothendieck regarding his discovery of the Galois action on dessins d'enfants: This discovery, which is technically so simple, made a very strong impression on me, and it represents a decisive turning point in the course of my reflections, a shift in particular of my centre of interest in mathematics, which suddenly found itself strongly focused.

This is surely because of the very familiar, non-technical nature of the objects considered, of which any child’s drawing scrawled on a bit of paper (at least if the drawing is made without lifting the pencil) gives a perfectly explicit example.

To such a dessin we find associated subtle arithmetic invariants, which are completely turned topsy-turvy as soon as we add one more stroke.Part of the theory had already been developed independently by Jones & Singerman (1978) some time before Grothendieck.

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.

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.

However, this construction identifies the Riemann surface only as a manifold with complex structure; it does not construct an embedding of this manifold as an algebraic curve in the complex projective plane, although such an embedding always exists.

can be used to define gluing instructions for a collection of halfspaces that together form a Riemann surface homeomorphic to

; mapping each halfspace by the identity to the Riemann sphere produces a Belyi function

that lead to combinatorially equivalent dessins d'enfants are biholomorphic, and Belyi's theorem implies that, for any compact Riemann surface

Under the interpretation of a clean dessin as a map, an arbitrary dessin is a hypermap: that is, a drawing of a hypergraph in which the black points represent vertices and the white points represent hyperedges.

The five Platonic solids – the regular tetrahedron, cube, octahedron, dodecahedron, and icosahedron – viewed as two-dimensional surfaces, have the property that any flag (a triple of a vertex, edge, and face that all meet each other) can be taken to any other flag by a symmetry of the surface.

If a regular map is used to generate a clean dessin, and the resulting dessin is used to generate a triangulated Riemann surface, then the edges of the triangles lie along lines of symmetry of the surface, and the reflections across those lines generate a symmetry group called a triangle group, for which the triangles form the fundamental domains.

For example, the figure shows the set of triangles generated in this way starting from a regular dodecahedron.

In this case, the starting surface is the quotient of the hyperbolic plane by a finite index subgroup Γ in this group.

Any embedding of a tree has a single region, and therefore by Euler's formula lies in a spherical surface.

The corresponding Belyi pair forms a transformation of the Riemann sphere that, if one places the pole at

as a Belyi function from the Riemann sphere to itself because its critical values all lie in the set

The corresponding dessin d'enfant is a star having one central black vertex connected to

The corresponding dessins take the form of path graphs, alternating between black and white vertices, with

Up to normalization and linear transformations of its argument, the Shabat polynomial is uniquely determined from a coloring of an embedded tree, but it is not always straightforward to find a Shabat polynomial that has a given embedded tree as its dessin d'enfant.

These functions, though closely related to each other, are not equivalent, as they are described by the two nonisomorphic trees shown in the figure.

More generally, due to the fact that the critical values of any Belyi function are the pure rationals 0, 1, and

contained in the stabilizer of the dessin, and the field of moduli of the orbit corresponds to the smallest normal extension of

For instance, for the two conjugate dessins considered in this section, the field of moduli of the orbit is