Dessin d'enfant

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In mathematics, a dessin d'enfant (French for a "child's drawing", plural dessins d'enfants, "children's drawings") is a type of graph drawing used by Grothendieck (1984) and later researchers to study Riemann surfaces and to provide combinatorial invariants for the action of the absolute Galois group of the rational numbers.

Combinatorially, a dessin d'enfant is simply a connected undirected graph with a cyclic ordering of edges at each vertex, such that each vertex is colored black or white with no edge having endpoints of the same color. That is, it is a rotation system describing an embedding of a bipartite graph onto an orientable surface, together with a choice of coloring for the graph. Any dessin provides the surface on which it is embedded with a structure as a Riemann surface; the absolute Galois group acts on these surfaces and thereby on the underlying dessins.

Zapponi (2003) quotes Grothendieck regarding his discovery of the Galois action on dessins d'enfants:

For a more detailed treatment of this subject, see Schneps (1994) or Lando & Zvonkin (2004).

[edit] Riemann surfaces and Belyi pairs

The dessin d'enfant arising from the rational function ƒ = −(x−1)³(x−9)/64x. Not to scale.

Transforming a dessin d'enfant into a gluing pattern for halfspaces of a Riemann surface by including points at infinity.

Consider^[1] the rational function

Viewed as a transformation of the Riemann sphere, ƒ has the following critical points and critical values:^[2]

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

We 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 (3 ± 2√3), and arcs at the preimages of the line segment [0,1]. 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, we may form a compact Riemann surface, and a map from that surface to the Riemann sphere, equivalent to the one we started with. 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) we 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(0) of 1, and its edges placed along the preimages ƒ^-1([0,1]) of the line segment [0,1]. 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.

[edit] Maps and hypermaps

The degree of a vertex in a dessin may equally be viewed as its algebraic degree as a critical point of the Belyi function, or as its graph-theoretic degree, the number of edges at the vertex. A dessin such as the one in the example above, in which all white points have degree two, corresponds to a map on the surface X, an embedding of a graph connecting the black points only in such a way that each face is a disk. Dessins with this property are known as clean, and correspond to pure Belyi functions. It is common to draw only the black points of a clean dessin and to leave the white points unmarked; one can recover the full dessin by adding a white point at the midpoint of each edge of the map. For instance, the dessin shown in the figure could be drawn more simply in this way as a pair of black points with an edge between them and a self-loop on one of the points. If a map corresponds to a Belyi function ƒ, its dual graph corresponds to the multiplicative inverse 1/ƒ.^[3]

Any dessin can be transformed into a clean dessin on the same surface, by recoloring all of its points as black and adding new white points on each of its edges. The corresponding transformation of Belyi pairs is to replace a Belyi function β by the pure Belyi function γ = 4β(β − 1). We may calculate the critical points of γ directly from this formula: γ⁻¹(0) = β⁻¹(0) ∪ β⁻¹(0), γ⁻¹(∞) = β⁻¹(∞), and γ⁻¹(1) = β⁻¹(1/2). Thus, γ⁻¹(1) is the preimage under β of the midpoint of the line segment [0,1], and the edges of the dessin formed from γ subdivide the edges of the dessin formed from β.

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.

[edit] Trees and Shabat polynomials

The dessin d'enfant corresponding to the sextic monomial p(x) = x⁶.

The Chebyshev polynomials and the corresponding dessins d'enfants, alternately-colored path graphs.

The simplest bipartite graphs are the trees. Any embedding of a tree has a single region, and therefore by Euler's formula lies on a spherical surface. The corresponding Belyi pair forms a transformation of the Riemann sphere which, if we place the pole at ∞, can be represented as a polynomial. Conversely, any polynomial with 0 and 1 as its finite critical values forms a Belyi function from the Riemann sphere to itself, having a single infinite-valued critical point, and corresponding to a dessin d'enfant that is a tree. The degree of the polynomial equals the number of edges in the corresponding tree.

For example, we may take p to be the monomial p(x) = x^d having only one finite critical point and critical value, both zero. Although 1 is not a critical value for p, we may still interpret it as a Belyi function from the Riemann sphere to itself. The corresponding dessin d'enfant is a star having one central black vertex connected to d white leaves (a complete bipartite graph K_1,d).

More generally, a polynomial p(x) having two critical values y₁ and y₂ is known as a Shabat polynomial, after George Shabat. Such a polynomial may be normalized into a Belyi function, with its critical values at 0 and 1, by the formula

but it may be more convenient to leave p in its un-normalized form.^[4]

An important family of examples of Shabat polynomials are given by the Chebyshev polynomials of the first kind, T_n(x), which have −1 and 1 as critical values. The corresponding dessins take the form of path graphs, alternating between black and white vertices, with n edges in the path. Due to the connection between Shabat polynomials and Chebyshev polynomials, Shabat polynomials themselves are sometimes called generalized Chebyshev polynomials.

Different trees will, in general, correspond to different Shabat polynomials, as will different embeddings or colorings of the same tree. 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.

[edit] The absolute Galois group and its invariants

Two conjugate dessins d'enfants

The polynomial

may be made into a Shabat polynomial by choosing

^[5]

The two choices of a lead to two Belyi functions ƒ₁ and ƒ₂. These functions, though closely related to each other, are not equivalent, as they are described by the two nonisomorphic trees shown in the figure.

However, as these polynomials are defined over the algebraic number field ℚ(√21), they may be transformed by the action of the absolute Galois group Γ of the rational numbers. An element of Γ that transforms √21 to -√21 will transform ƒ₁ into ƒ₂ and vice versa, and thus can also be said to transform each of the two trees shown in the figure into the other tree. More generally, due to the fact that the critical values of any Belyi function are the pure rationals 0, 1, and ∞, these critical values are unchanged by the Galois action, so this action takes Belyi pairs to other Belyi pairs. We may define an action of Γ on any dessin d'enfant by the corresponding action on Belyi pairs; this action, for instance, permutes the two trees shown in the figure.

Due to Belyi's theorem, the action of Γ on dessins is faithful,^[6] so the study of dessins d'enfants can tell us much about Γ itself. In this light, it is of great interest to understand which dessins may be transformed into each other by the action of Γ and which may not. For instance, one may observe that the two trees shown have the same degree sequences for their black nodes and white nodes: both have a black node with degree three, two black nodes with degree two, two white nodes with degree two, and three white nodes with degree one. This equality is not a coincidence: whenever Γ transforms one dessin into another, both will have the same degree sequence. The degree sequence is one known invariant of the Galois action, but not the only invariant.

The stabilizer of a dessin is the subgroup of Γ consisting of group elements that leave the dessin unchanged. Due to the Galois correspondence between subgroups of Γ and algebraic number fields, the stabilizer corresponds to a field, the field of moduli of the dessin. An orbit of a dessin is the set of all other dessins into which it may be transformed; due to the degree invariant, orbits are necessarily finite and stabilizers are of finite index. We may similarly define the stabilizer of an orbit and the corresponding field of moduli of the orbit, another invariant of the dessin. The stabilizer of the orbit is the maximal normal subgroup of Γ contained in the stabilizer of the dessin, and the field of moduli of the orbit corresponds to the smallest normal extension of ℚ that contains the field of moduli of the dessin. For instance, for the two conjugate dessins considered in this section, the field of moduli of the orbit is ℚ(√21). The two Belyi functions ƒ₁ and ƒ₂ of this example are defined over the field of moduli, but there exist dessins for which the field of definition of the Belyi function must be larger than the field of moduli.^[7]

[edit] Notes

[edit] References

• Grothendieck, A. (1984), Esquisse d'un programme, <http://www.institut.math.jussieu.fr/~leila/grothendieckcircle/EsquisseFr.pdf> ; English translation.

• Lando, Sergei K. & Zvonkin, Alexander K. (2004), Graphs on Surfaces and Their Applications, vol. 141, Encyclopaedia of Mathematical Sciences: Lower-Dimensional Topology II, Berlin, New York: Springer-Verlag, ISBN 978-3-540-00203-1 . See especially chapter 2, "Dessins d'Enfants", pp. 79–153.

• Schneps, Leila (1994), The Grothendieck Theory of Dessins d'Enfants, London Mathematical Society Lecture Note Series, Cambridge: Cambridge University Press, ISBN 978-0-521-47821-2 .

• Zapponi, Leonardo (August 2003), “What is a Dessin d'Enfant”, Notices of the American Mathematical Society 50: 788–789, ISSN 0002-9920, <http://www.ams.org/notices/200307/what-is.pdf> .