Regular Dessins Research Articles

Almost Abelian regular dessins d'enfants

A regular dessin d’enfant, in this paper, will be a pair (S , β), where S is a closed Riemann surface and β : S → Ĉ is a regular branched cover whose branch values are contained in the set {∞, 0, 1}. Let Aut(S , β) be the group of automorphisms of (S , β), that is, the deck group of β. If Aut(S , β) is Abelian, then it is known that (S , β) can be defined over the field of rational numbers Q. In this paper we prove that, if A is an Abelian group and Aut(S , β) A ⋊ Z2, then (S , β) is also definable over Q. Moreover, if A Zn, then we provide explicitly these dessins over Q.

Hypermaps and multiply quasiplatonic Riemann surfaces

Generalising an example by Girondo and Wolfart, we use finite group theory to construct Riemann surfaces admitting two or more regular dessins (i.e. orientably regular hypermaps) with automorphism groups of the same order, and in many cases with isomorphic automorphism groups.

Wilson's map operations on regular dessins and cyclotomic fields of definition

Dessins d'enfants can be seen as bipartite graphs embedded in compact orientable surfaces. According to Grothendieck and others, a dessin uniquely determines a complex structure on the surface, even an algebraic structure as a projective algebraic curve defined over a number field. Combinatorial properties of the dessin should therefore determine the equations and also structural properties of the curve, such as the field of moduli or the field of definition. However, apart from a few series of examples, very few general results concerning such correspondences are known. As a step in this direction, we present a graph theoretic characterisation of certain quasiplatonic curves defined over cyclotomic fields, based on Wilson's operations on maps: these leave invariant the graph but change the cyclic ordering of edges around the vertices, therefore they change the embeddings, and hence the dessins, and hence the conformal and algebraic structure of the underlying curves. Under suitable assumptions, satisfied by many series of regular dessins, these changes coincide with the effect of Galois conjugation. This coincidence allows one to draw conclusions about Galois orbits and fields of definition of dessins. The possibilities afforded by these techniques, and their limitations, are illustrated by a new look at some known examples and a study of dessins based on the regular embeddings of complete graphs.

How many quasiplatonic surfaces?

We show that the number of isomorphism classes of quasiplatonic Riemann surfaces of genus ≦ g has a growth of type $$g^{{\log g}} .$$ The number of non-isomorphic regular dessins of genus ≦ g has the same growth type.

Conjugators of Fuchsian groups and quasiplatonic surfaces

Let G be a Fuchsian group containing two torsion free subgroups defining isomorphic Riemann surfaces. Then these surface subgroups K and αKα−1 are conjugate in PSL(2,R), but in general the conjugating element α cannot be taken in G or a finite index Fuchsian extension of G. We will show that in the case of a normal inclusion in a triangle group G these α can be chosen in some triangle group extending G. It turns out that the method leading to this result allows also to answer the question of how many different regular dessins of the same type can exist on a given quasiplatonic Riemann surface.

Multiply Quasiplatonic Riemann Surfaces

The aim of this article is the study of the circumstances under which a compact Riemann surface may contain two regular dessin d'enfants of different types. In terms of Fuchsian groups, an equivalent condition is the uniformizing group being normally contained in several different triangle groups. The question is answered in a graph-theoretical way, providing algorithms that decide if a surface that carries a regular dessin (a quasiplatonic surface) can also carry other regular dessins. The multiply quasiplatonic surfaces are then studied depending on their arithmetic character. Finally, the surfaces of lowest genus carrying a large number of nonarithmetic regular dessins are computed.

Characters and galois invariants of regular dessins

We described a new invariant for the action of the absolute Galois groups on the set of Grothendieck dessins. It uses the fact that the automorphism groups of regular dessins are isomorphic to automorphism groups of the corresponding Riemman surfaces and define linear represenatations of the space of holomorphic differentials. The characters of these representations give more precise information about the action of the Galois group than all previously known invariants, as it is shown by a series of examples. These examples have their own interest because the Galois action on them is described only using properties of the fixed points in the canonical model, without the explicit knowledge of equations.