Dessins d'Enfants on Riemann Surfaces

  The term "dessin d'enfant" (children's drawing) was coined by Alexander Grotendieck (1928-2014) in the 1980's because he was so enthusiastic about the simplicity of the graphs that represented very complex mathematical objects. The graphs may be simple, but to understand the mathematics that they represent is not. So, unless you are familiar with the concept and with algebraic geometry, it might be a somewhat misleading title.

The text of this monograph grew out of notes taken while the authors were lecturing on this topic. Thus there is a definition of dessins (there are actually three different but equivalent definitions). The necessary (background) material is either briefly reviewed or added in appendices after the relevant chapters. It is however necessary that the reader is familiar with complex functions and a thorough knowledge of group theory. This is not the best place to become familiar with algebraic geometry. Knowledge of graphs and Galois theory might also help. Thus the book is intended for properly trained mathematics graduate students or researchers. The subject is fascinating though and it is connecting several different mathematical disciplines (complex functions, graphs, group theory, topology, Galois theory,...) which makes this domain all the more interesting to work in.

For the reader familiar with the subject, the book will bundle the material in a nice way, and it will probably suffice to say that the text consists of three parts: (I) giving an introduction to the concepts that play a major role and a survey of the results obtained in the last 40 years; (II) is a study of regular dessins, how they can be constructed and their classification; (III) discussed two applications: an abc-like theorem for the degrees of functions defining algebraic curves and Beauville surfaces.

For those not familiar with the subject, it requires some more explanation. A complex smooth meromorphic function β(z) defined on a (compact oriented) Riemann surface X (you can think of the Riemann sphere for simplicity) and taking values on the Riemann sphere is a Belyĭ function if it has no critical points outside {0,1,∞}, i.e., these correspond to branch or ramification points of a certain order. So there are in general several preimages of these points. Taking the preimages of the interval [0,1] results in a bipartite graph. The preimages of 0 are colored white (they are the zeros of β), those of 1 are colored black (these are the zeros of 1-β) and the preimages of the interval define edges, each connecting a white and a black point. This graph is a dessin (d'enfants). The preimages of ∞ are left out but if we had included them, there would be one in each of the connected regions defined by the edges of the dessin. If we color these red, then we can generate a triangulation by connecting the red vertices with the black and white ones on the boundary of its region. A triangle face is the preimage of the lower half plane if its boundary has white, black, red vertices in clockwise order, otherwise it is the preimage of the upper half plane (which is a triangulation of the Riemann sphere). The Belyĭ functions are just special cases of more general smooth functions, but transformations can arrange that in the end one has to study mostly those.

Of course the situation can be much more complicated because Riemann surfaces can have a nonzero genus, or one may want to work with projective coordinates because we are dealing with points at infinity, and the algebraic curves that the functions define can be very complex, etc. On the other hand, there are many different tools to deal with the problem because there are also different ways to define a dessin. There is the connection with groups, which can be explained as follows. The same triangulation we have described for β can be described by functions 1/β, 1-β, 1/(1-β), 1-1/β, and β/(1-β), which just give permutations of the colors of the vertices, and hence a permutation of the edges. Two permutations (the white and black vertices characterize their orbits) generate a transitive group (called the monodromy group). This is a quotient group of a triangle group and the group of isometries of the hyperbolic plane. So that introduces group theory into the picture. Not surprisingly taking into account the nice symmetries of the pictures of the triangulations or dessins.

The dessins are useful ways to study Belyĭ functions, but more general hypergraphs introduced by Cori in the context of computer graphics is even more appropriate (although these are not the main topic in this book). Belyĭ (1951-2001) was interested in inverse Galois theory (i.e., find information on the absolute Galois group of automorphisms of the closed field of algebraic numbers). The important Belyĭ theorem (1979) says that it is sufficient to study Belyĭ functions (hence dessins) to understand algebraic curves and the corresponding compact Riemann surfaces.

All these connections and much more is explained in part I. Part II is a discussion of regular dessins, which are dessins with the highest possible symmetry. Analyzing the symmetry requires an intensive use of group theory, which I will not elaborate on here. For genus 0 (Riemann sphere) or genus 1 (torus) there are infinitely many symmetries possible, but for a genus larger than 1, the symmetries are limited. The Hurwitz bound says that the number of automorphisms of compact Riemann surfaces of genus g is bounded by 84(g-1) and the quasiplatonic curves of genera 2,3, and 4 are analyzed in detail. What is said about dessins can be generalized to maps, which are characterized by a compact Riemann surface X, a graph $\cal G$ embedded in X and an automorphism group G. Classification can be done on the basis of each of these, of course taking the genus into account.

In part III we find an application in the abc conjecture (suggests an upper bound on the the size of the integers a, b and c solving a+b+c=0 in terms of their prime divisors). There is an analogy with the degrees of self covers of prime degree of an elliptic curve defined over a number field that has complex multiplication. The optimal solutions turn out to be exactly the Belyĭ functions. The extension from curves to complex surfaces is a considerable complication. The Beauville surfaces however allow such an analysis, and this is studied in the last chapter of the book.

It is clear from this brief survey that this is a book for the specialists. Theorems are formulated, but proofs are often rather short, or are only outlined. The text is also regularly interrupted by formulations of exercises which ask to prove some intermediate result or apply a definition in a particular situation (some brief hints are added at the end of the book). The book can thus be used as a textbook for a course on this fascinating topic. With a proper preparation, it gives a good entry point to current research. Not the most general situation is always described, but all chapters are completed with many references with pointers in the text that may refer to a more extensive introduction or survey on a particular topic or to more advanced treatments or a connection that is not elaborated in this book.

Reviewer: 
Adhemar Bultheel
Book details

This is a book based on lecture notes for courses given by the authors. They deal with the study of bipartite graphs on Riemann surfaces known as dessins d'enfants, a term coined by A. Grotendieck. These are simple characterizations of certain maps on Riemann surfaces. Knowledge of complex functions, Riemann surfaces and group theory are prerequisites. The book has 3 parts. The first one gives the necessary definitions and recalls preliminaries, the second studies regular dessins for low genera surfaces, having a maximal symmetry and the last part gives an analog of the abc conjecture. It concerns a bound for degrees of Belyĭ functions associated with algebraic curves and surfaces.

Author:  Publisher: 
Published: 
2016
ISBN: 
978-3-319-25566-8 (hbk)
Price: 
90.09 € (hbk)
Pages: 
273
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