Multiply Quasiplatonic Riemann Surfaces

compact Riemann surfaces, e.g., two-dimensional compact manifolds with complex analytic structure, have been studied from very different points of view. On the one hand, they arise out of complex algebraic curves, and on the other hand as quotient spaces by the action of groups of Möbius transformations. These two settings are related in a highly obscure way, but the connection can sometimes be shown more explicitly, as in the case of Belyi surfaces. We shall just provide a brief introduction to them in this section, but the interested reader can find the details in [Cohen et al. 94] or [Jones and Singerman 96] and in the references given there. Note that Jones and Singerman often use the language of hypermaps instead of that of dessins that we employ here. In the abstract setting, a Belyi surface is defined to be a compact Riemann surface X for which a holomorphic function β : X → Ĉ with at most three branch values can be defined. Such a β is called a Belyi function, and the branch values can be supposed to be contained in {0, 1,∞} after normalization. The following famous result makes clear why this class of surfaces is so interesting (for the proof, see [Belyi 80] or [Wolfart 97]). Theorem 2.1. (Belyi.) X is a Belyi surface if and only if the corresponding algebraic curve can be defined over Q̄. Suppose β : X → Ĉ is a Belyi function. We associate to β an embedded graph in X by considering β−1{t ∈ R | 0 ≤ t ≤ 1}. This is a bipartite graph (its vertices being β−1{0, 1}), since we can colour the preimages of 0 in black, and the preimages of 1 in white, and then every two adjacent vertices have different color. This motivates the following: Definition 2.2. A dessin d’enfant is a bipartite graph D embedded in a compact Riemann surface X, such that each component of X D is simply connected. Those components are called the faces of D. The combinatorial structure of a given dessin D can be encoded in the following way: Label the edges of D with numbers 1, 2, . . . , N . Now, if a black vertex is fixed, several edges are adjacent to it, and the anticlockwise orientation of the surface gives a cyclic permutation of them. Hence, if D contains B black vertices, we get a permutation rb that is a product of B disjoint cycles, and the length of each cycle is the valency of the corresponding black vertex. In the same way, we construct a permutation rw looking at the white vertices, and we find that the cycles of the permutation rf = (rwrb)−1 give information about the faces, since every cycle describes half the edges going around a face. Thus, a cycle of length k of rf corresponds to a 2k-gonal face. We say that D is of type (l,m, n) if l (respectively, m) is the least common multiple of the valencies of the black vertices (respectively, the white vertices), and n is half the least common multiple of the face valencies. These are, of course, just the orders of rb, rw, and rf . The subgroup GD of SN generated by these three permutations is called the monodromy group of the dessin. It is not difficult to reconstruct the dessin from its monodromy, since rb, rw, and rf carry all the combinatorial information. In fact, the complex structure of a Belyi surface is determined by its dessin. More precisely, the combinatoric data of the dessin determines a Fuchsian group that gives as quotient space the Belyi surface. Given the integers l,m, and n, let T (l,m, n) be a hyperbolic triangle from angles π/l, π/m, and π/n. Construct the group ∆̃(l,m, n) generated by the reflections across the three sides of T (l,m, n). Let ∆(l,m, n) be the index two subgroup formed by the orientation preserving elements of ∆̃(l,m, n) (these are the words of even length in the three reflections). The Fuchsian group ∆(l,m, n) is called a triangle group, and it has the well-known presentation < γb, γw, γf ; γ b = γ m w = γ n f = γfγwγb = 1 > (the three generators giving this presentation can be chosen in some geometrical way, as explained at the beginning of Section 4.1). There is a natural group homomorphism determined by every dessinD of type (l,m, n), going from the triangle group ∆(l,m, n) onto some group of permutations, its definition being simply θ : ∆(l,m, n) −→ GD γi −→ ri