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. The name of these embeddings is French for a "child's drawing"; its plural is either dessins d'enfant, "child's drawings", or dessins d'enfants, "children's drawings".
A dessin d'enfant is a
Any dessin can provide the surface it is embedded in with a structure as a Riemann surface. It is natural to ask which Riemann surfaces arise in this way. The answer is provided by Belyi's theorem, which states that the Riemann surfaces that can be described by dessins are precisely those that can be defined as algebraic curves over the field of algebraic numbers. The absolute Galois group transforms these particular curves into each other, and thereby also transforms the underlying dessins.
For a more detailed treatment of this subject, see Schneps (1994) or Lando & Zvonkin (2004).
History
19th century
Early proto-forms of dessins d'enfants appeared as early as 1856 in the icosian calculus of William Rowan Hamilton;[1] in modern terms, these are Hamiltonian paths on the icosahedral graph.
Recognizable modern dessins d'enfants and
20th century
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. I do not believe that a mathematical fact has ever struck me quite so strongly as this one, nor had a comparable psychological impact. 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. They outline the correspondence between maps on topological surfaces, maps on Riemann surfaces, and groups with certain distinguished generators, but do not consider the Galois action. Their notion of a map corresponds to a particular instance of a dessin d'enfant. Later work by Bryant & Singerman (1985) extends the treatment to surfaces with a boundary.
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 where and are polynomials, transforms the Riemann sphere by mapping it to itself. Consider, for example,[6] 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 -to-one way onto its image. The number is known as the degree of the critical point and the transformed image of a critical point is known as a
critical point x | critical value f(x) | degree |
---|---|---|
One may form a dessin d'enfant from by placing black points at the
In the other direction, from this dessin, described as a combinatorial object without specifying the locations of the critical points, one may form a
The same construction applies more generally when is any Riemann surface and is a Belyi function; that is, a holomorphic function from to the Riemann sphere having only 0, 1, and as critical values. A pair of this type is known as a Belyi pair. From any Belyi pair one can form a dessin d'enfant, drawn on the surface , that has its black points at the preimages of 0, its white points at the preimages of 1, and its edges placed along the preimages of the line segment . Conversely, any dessin d'enfant on any surface 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 on , and therefore leads to a Belyi pair . Any two Belyi pairs that lead to combinatorially equivalent dessins d'enfants are biholomorphic, and Belyi's theorem implies that, for any compact Riemann surface defined over the algebraic numbers, there are a Belyi function and a dessin d'enfant that provides a combinatorial description of both and .
Maps and hypermaps
A vertex in a dessin has a graph-theoretic
Thus, any embedding of a graph in a surface in which each face is a disk (that is, a topological map) gives rise to a dessin by treating the graph vertices as black points of a dessin, and placing white points at the midpoint of each embedded graph edge. If a map corresponds to a Belyi function , its dual map (the dessin formed from the preimages of the line segment ) corresponds to the multiplicative inverse .[7]
A dessin that is not clean can be transformed into a clean dessin in 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 by the pure Belyi function . One may calculate the critical points of directly from this formula: , , and . Thus, is the preimage under of the midpoint of the line segment , and the edges of the dessin formed from subdivide the edges of the dessin formed from .
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.
Regular maps and triangle groups
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. More generally, a map embedded in a surface with the same property, that any flag can be transformed to any other flag by a symmetry, is called a regular map.
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
Conversely, given a Riemann surface that is a quotient of a tiling (a tiling of the sphere, Euclidean plane, or hyperbolic plane by triangles with angles , , and , the associated dessin is the Cayley graph given by the order two and order three generators of the group, or equivalently, the tiling of the same surface by -gons meeting three per vertex. Vertices of this tiling give black dots of the dessin, centers of edges give white dots, and centers of faces give the points over infinity.
Trees and Shabat polynomials
The simplest bipartite graphs are the trees. 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 , 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. Such a polynomial Belyi function is known as a Shabat polynomial,[8] after George Shabat.
For example, take to be the monomial having only one finite critical point and critical value, both
More generally, a polynomial having two critical values and may be termed a Shabat polynomial. Such a polynomial may be normalized into a Belyi function, with its critical values at 0 and 1, by the formula
An important family of examples of Shabat polynomials are given by the Chebyshev polynomials of the first kind, , which have −1 and 1 as critical values. The corresponding dessins take the form of path graphs, alternating between black and white vertices, with edges in the path. Due to the connection between Shabat polynomials and Chebyshev polynomials, Shabat polynomials themselves are sometimes called generalized Chebyshev polynomials.[9][10]
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.
The absolute Galois group and its invariants
The polynomial
However, as these polynomials are defined over the algebraic number field , they may be transformed by the
Due to Belyi's theorem, the action of on dessins is
The stabilizer of a dessin is the subgroup of consisting of group elements that leave the dessin unchanged. Due to the Galois correspondence between subgroups of 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. One may similarly define the stabilizer of an orbit (the subgroup that fixes all elements of the 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 contained in the stabilizer of the dessin, and the field of moduli of the orbit corresponds to the smallest normal extension of 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 . The two Belyi functions and 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.[13]
Notes
- ^ Hamilton (1856). See also Jones (1995).
- ^ Klein (1879).
- ^ le Bruyn (2008).
- ^ Klein (1878–1879a); Klein (1878–1879b).
- ^ Grothendieck (1984)
- ^ This example was suggested by Lando & Zvonkin (2004), pp. 109–110.
- ^ Lando & Zvonkin (2004), pp. 120–121.
- ^ Girondo & González-Diez (2012) p. 252
- ^ a b Lando & Zvonkin (2004), p. 82.
- Zbl 0898.14012
- ^ Lando & Zvonkin (2004), pp. 90–91. For the purposes of this example, ignore the parasitic solution .
- ^ acts faithfully even when restricted to dessins that are trees; see Lando & Zvonkin (2004), Theorem 2.4.15, pp. 125–126.
- ^ Lando & Zvonkin (2004), pp. 122–123.
References
- le Bruyn, Lieven (2008), Klein's dessins d'enfant and the buckyball.
- Bryant, Robin P.; Singerman, David (1985), "Foundations of the theory of maps on surfaces with boundary", MR 0780347.
- Girondo, Ernesto; González-Diez, Gabino (2012), Introduction to compact Riemann surfaces and dessins d'enfants, London Mathematical Society Student Texts, vol. 79, Cambridge: Zbl 1253.30001.
- Grothendieck, A. (1984), Esquisse d'un programme
- Hamilton, W. R. (17 October 1856), Letter to John T. Graves "On the Icosian". Collected in Halberstam, H.; Ingram, R. E., eds. (1967), Mathematical papers, Vol. III, Algebra, Cambridge: Cambridge University Press, pp. 612–625.
- Jones, Gareth (1995), "Dessins d'enfants: bipartite maps and Galois groups", Séminaire Lotharingien de Combinatoire, B35d: 4, archived from the original on 8 April 2017, retrieved 2 June 2010.
- Jones, Gareth; Singerman, David (1978), "Theory of maps on orientable surfaces", .
- S2CID 121056952, archived from the originalon 19 July 2011, retrieved 2 June 2010.
- S2CID 121407539, archived from the originalon 24 February 2012, retrieved 9 July 2010.
- S2CID 120316938, collected as pp. 140–165 in Oeuvres, Tome 3 Archived 19 July 2011 at the Wayback Machine.
- Lando, Sergei K.; Zvonkin, Alexander K. (2004), Graphs on Surfaces and Their Applications, Encyclopaedia of Mathematical Sciences: Lower-Dimensional Topology II, vol. 141, Berlin, New York: Zbl 1040.05001. See especially chapter 2, "Dessins d'Enfants", pp. 79–153.
- ISBN 978-0-521-47821-2.
- Zbl 0868.00040.
- Shabat, G.B.; Zbl 0790.14026.
- Singerman, David; Syddall, Robert I. (2003), "The Riemann Surface of a Uniform Dessin", Beiträge zur Algebra und Geometrie, 44 (2): 413–430, Zbl 1064.14030.
- Zapponi, Leonardo (August 2003), "What is a Dessin d'Enfant" (PDF), Zbl 1211.14001.