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## Über dieses Buch

This volume provides an introduction to dessins d'enfants and embeddings of bipartite graphs in compact Riemann surfaces. The first part of the book presents basic material, guiding the reader through the current field of research. A key point of the second part is the interplay between the automorphism groups of dessins and their Riemann surfaces, and the action of the absolute Galois group on dessins and their algebraic curves. It concludes by showing the links between the theory of dessins and other areas of arithmetic and geometry, such as the abc conjecture, complex multiplication and Beauville surfaces.
Dessins d'Enfants on Riemann Surfaces will appeal to graduate students and all mathematicians interested in maps, hypermaps, Riemann surfaces, geometric group actions, and arithmetic.

## Inhaltsverzeichnis

### Chapter 1. Historical and Introductory Background

Abstract
This chapter begins with a brief historical introduction to the theory of dessins d’enfants, from the early discovery of the platonic solids, through nineteenth-century work on Riemann surfaces, algebraic curves and holomorphic functions, and twentieth-century research on regular maps, to the fundamental and far-reaching ideas circulated by Grothendieck in the 1980s, and subsequent efforts to implement his programme. After this we summarise the background knowledge we will assume, together with suggestions for further reading. The second section gives a brief introduction to compact Riemann surfaces, including the Riemann-Hurwitz formula for the genus of a surface, and the equivalence of the categories of compact Riemann surfaces and of smooth complex projective algebraic curves. Elliptic curves (Riemann surfaces of genus 1) are treated in detail, as simple examples of subtler phenomena encountered later. The third section contains technical results on the existence of meromorphic functions with specific properties. In the final section we define Belyĭ functions and prove one direction of Belyĭ’s theorem, that such functions characterise algebraic curves defined over number fields, by using an algorithm which constructs a Belyĭ function on such a curve. We give a first definition of dessins d’enfants as the pre-images of the unit real interval [0, 1] under Belyĭ functions, and we discuss several simple examples of dessins.
Gareth A. Jones, Jürgen Wolfart

### Chapter 2. Graph Embeddings

Abstract
In this chapter we introduce maps and hypermaps, firstly as topological structures on surfaces, and then as equivalent algebraic objects, described by means of their monodromy groups. We give two further topological and group theoretic definitions of dessins, equivalent to that given in Chap. 1 in terms of Belyĭ functions; these definitions involve bipartite graphs embedded in surfaces, and 2-generator permutation groups. Morphisms, automorphisms and quotients of maps and dessins are defined, together with their regularity properties. Various other possibilities for the graphical representation of dessins are briefly discussed. The chapter includes an instructive example, in which a very simple dessin gives rise to a group of order 95040, the Mathieu group M 12 . The chapter closes with a summary of the finite simple groups, which appear first here and then again in later chapters of this book.
Gareth A. Jones, Jürgen Wolfart

### Chapter 3. Dessins and Triangle Groups

Abstract
The first section of this chapter gives a short account of uniformisation theory for Riemann surfaces, in which the main result is the extended Riemann mapping theorem. We describe the classification of cocompact Fuchsian groups, and the local and global properties of quotients of the hyperbolic plane by such groups. We discuss the inclusion relations between these groups, classified by Singerman, giving particular attention to triangle groups. These groups turn out to be very important, not only for a better understanding of maps and hypermaps, but also because certain automorphic functions related to them act as a source for all Belyĭ functions. These ideas are illustrated with Klein’s quartic curve, a classic example of a Riemann surface of genus 3. In the second section we use triangle groups to prove Grothendieck’s important observation, that even a purely topological definition of a dessin as a graph embedded in a compact oriented surface leads to a unique conformal structure on this surface, so that the dessin corresponds to a Belyĭ function on the resulting Riemann surface. This shows that our previous definitions of dessins (via Belyĭ functions, embedded graphs or permutations) are in fact equivalent. Grothendieck attributed this result to Malgoire and Voisin, and in the meantime there have been many proofs, such as that given by Voevodsky and Shabat, but here we use ideas of Singerman from pre-dessins times, involving maps and triangle groups. The chapter includes an appendix explaining group presentations.
Gareth A. Jones, Jürgen Wolfart

### Chapter 4. Galois Actions

Abstract
This chapter first collects basic material about Galois theory for finite and infinite field extensions, with examples chosen from number fields and function fields. The latter examples provide a link between Galois groups and covering groups for regular coverings. Another important example is the absolute Galois group $$\mathbb{G}$$, the automorphism group of the field of all algebraic numbers: as the projective limit of the (finite) Galois groups of the Galois extensions of the rationals, this is a profinite group, with a natural topology, the Krull topology, making it a topological group. Belyĭ’s Theorem implies that $$\mathbb{G}$$ has a natural action on dessins, through its action on the algebraic numbers defining them. As observed by Grothendieck, this action is faithful, so it gives a useful insight into the Galois theory of algebraic number fields. In the second section, moduli fields of algebraic curves are defined, and we discuss their relation to fields of definition. Weil’s cocycle condition is explained. We sketch two proofs of the other direction of Belyĭ’s theorem, that a curve can be defined over an algebraic number field if it admits a Belyĭ function. We list some Galois invariants and non-invariants of dessins, which are useful in determining orbits of $$\mathbb{G}$$, and we give a proof due to Lenstra and Schneps that $$\mathbb{G}$$ acts faithfully on the set of dessins formed from trees in the plane.
Gareth A. Jones, Jürgen Wolfart

### Chapter 5. Quasiplatonic Surfaces, and Automorphisms

Abstract
Quasiplatonic Riemann surfaces or algebraic curves, sometimes also called curves with many automorphisms or triangle curves, can be characterised in many equivalent ways, for example as those curves having a regular dessin, one with the greatest possible degree of symmetry. The sphere and the torus each support infinitely many regular dessins, easily described in both cases. For each genus g > 1 there are, up to isomorphism, only finitely many regular dessins; this chapter gives complete lists for genera 2, 3 and 4, and discusses methods for counting and classifying them. These methods often involve counting generating triples for finite groups, in some cases with the aid of character theory (which we briefly summarise) and Möbius inversion. We present several important infinite families of quasiplatonic curves, such as Hurwitz and Macbeath-Hurwitz curves, Lefschetz and Accola-Maclachlan curves. We prove that like their counterparts, the curves with trivial automorphism group, quasiplatonic curves can be defined over their field of moduli. Many of the automorphism groups appearing in this chapter are 2-dimensional linear or projective groups over finite fields, so we summarise their most relevant properties in the final section.
Gareth A. Jones, Jürgen Wolfart

### Chapter 6. Regular Maps

Abstract
Regular maps can be considered as special types of regular dessins. They include some of the oldest geometric objects known to mankind, in the form of the Platonic solids. These, together with the dihedra and their duals, the hosohedra (meaning ‘with many faces’), are the regular maps on the sphere. In this chapter we study their generalisations to maps on compact Riemann surfaces of arbitrary genus. We show how to classify regular maps in terms of their genus, paying particular attention to the cases of genus 0—as above—and of genus 1, where there are infinitely many regular maps, associated with ideals in the rings of Gaussian and Eisenstein integers. For each genus g ≥ 2 the Hurwitz bound implies that there are only finitely many regular maps, and we briefly consider the classification for g = 2. We also outline how one can classify regular maps in terms of their automorphism group, and we consider which groups can arise in this context.
Gareth A. Jones, Jürgen Wolfart

### Chapter 7. Regular Embeddings of Complete Graphs

Abstract
The methods described for classifying regular maps in terms of their genus or automorphism group can be extended, without much difficulty, to all regular dessins. However, the methods for classifying regular maps in terms of their embedded graphs are much more specific to maps, and do not extend so easily to other dessins. Gardiner, Nedela, Širáň and Škoviera developed a general strategy for this problem, involving the search for subgroups of the automorphism group of the graph which act transitively on the vertices, such that the stabiliser of a vertex is a cyclic group acting regularly on its neighbours. To illustrate the general methods available, we concentrate on a rather simple class of graphs, namely the complete graphs K n . Using results about a class of permutation groups called Frobenius groups, we show that their only regular embeddings are those constructed by Biggs, using Cayley graphs for the additive groups of finite fields. We enumerate such maps, and show that their automorphism groups are 1-dimensional affine groups. We also show how these maps are related to cyclotomic polynomials, and to primitive polynomials over finite fields.
Gareth A. Jones, Jürgen Wolfart

### Chapter 8. Wilson Operations

Abstract
As shown in Chap. 3, a dessin uniquely determines all the relevant properties of its underlying Belyĭ surface. A key invariant is the moduli field of the corresponding algebraic curve, and a major step in determining this is to understand the action of the absolute Galois group $$\mathbb{G}$$ on regular dessins. Under relatively mild conditions this action can be described combinatorially with some map and hypermap operations, the so-called Wilson (hole) operations, introduced around the same time as Belyĭ functions. However, their role in the understanding of Galois actions on dessins has only recently been discovered. We give several examples, based on the regular embeddings of complete graphs classified in Chap. 7 In the final section we consider the group of all operations on dessins, introduced by James, showing that it is isomorphic to the outer automorphism group of the free group of rank 2, and hence to $$\mathop{\mathrm{GL}}\nolimits _{2}(\mathbb{Z})$$. As an example we consider the action of this group on the 19 regular dessins with automorphism group A 5.
Gareth A. Jones, Jürgen Wolfart

### Chapter 9. Further Examples

Abstract
In the first part of this chapter we outline the classification of the regular embeddings of two families of regular graphs, namely certain generalised Paley graphs (using Cayley maps for finite fields), and the complete bipartite graphs K n, n . We describe their automorphism groups, and characterise the generalised Paley maps as those maps for which the automorphism group acts primitively and faithfully on the vertices. In the case of the complete bipartite graphs, results of Huppert, Itô and Wielandt on factorisations of groups, and of Hall on solvable groups, are used in the classification. We show how Wilson operations act on these two families of maps, and we use this to investigate their Galois orbits and fields of definition. In the second part of this chapter we extend the action of Wilson operations from regular maps to regular dessins, concentrating on those dessins which embed a complete bipartite graph K p, q where p and q are distinct primes. Under suitably favourable conditions on p and q, we can classify these dessins and use Wilson operations to determine their Galois orbits and fields of definition. Finally, we determine explicit equations for the associated quasiplatonic curves, a problem which is completely intractable in most other cases.
Gareth A. Jones, Jürgen Wolfart

### Chapter 10. Arithmetic Aspects

Abstract
In this chapter we discuss two links between dessins and arithmetic. The abc problem for integers concerns conjectured bounds on the size of integers a, b, c satisfying $$a + b + c = 0$$, in terms of the primes dividing them. Closely related to some of the deepest theorems and most difficult conjectures in number theory, it is a major open problem. However, its analogue for meromorphic functions on compact Riemann surfaces is comparatively easy to solve. In this case, Belyĭ functions provide extremal examples which show that the inequality in the main result is sharp. In the second section, we prove a dessin-theoretic criterion for elliptic curves (those of genus 1) to have complex multiplication, that is, to have non-trivial endomorphisms. This criterion relies on a very special property of curves and dessins of genus 1, that they have infinitely many unramified self-coverings.
Gareth A. Jones, Jürgen Wolfart

### Chapter 11. Beauville Surfaces

Abstract
Beauville surfaces are examples of complex surfaces, that is, they are 2-dimensional analogues of the complex algebraic curves (1-dimensional over $$\mathbb{C}$$) which we have considered so far. They are formed from certain pairs of regular dessins with the same automorphism group G (called a Beauville group) acting freely on their product. In the first section of the chapter we describe the basic theory of Beauville surfaces, in particular their construction by means of quasiplatonic (triangle) curves. We study Beauville’s original example, and we classify and enumerate its generalisations, based on Fermat curves, in which G is an abelian group. In the second section we consider which members of various other families of finite groups can be Beauville groups: these include symmetric groups, alternating groups and 2-dimensional projective linear groups over finite fields. In the third and final section we discuss topological invariants of a Beauville surface, such as its Euler characteristic, fundamental group and automorphism group, and their behaviour under Galois actions. We consider conditions under which a Beauville surface has a real model, and we give examples of Serre’s observation that Galois conjugation can change the topology of a complex variety.
Gareth A. Jones, Jürgen Wolfart

### Backmatter

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