Introduction to Compact Riemann Surfaces and Dessins d’Enfants

Abstract

Few books on the subject of Riemann surfaces cover the relatively modern theory of dessins d'enfants (children's drawings), which was launched by Grothendieck in the 1980s and is now an active field of research. In this 2011 book, the authors begin with an elementary account of the theory of compact Riemann surfaces viewed as algebraic curves and as quotients of the hyperbolic plane by the action of Fuchsian groups of finite type. They then use this knowledge to introduce the reader to the theory of dessins d'enfants and its connection with algebraic curves defined over number fields. A large number of worked examples are provided to aid understanding, so no experience beyond the undergraduate level is required. Readers without any previous knowledge of the field of dessins d'enfants are taken rapidly to the forefront of current research.

A panaroma of the fundamental group of the modular orbifold

“…We refer the interested reader to [9] for further reading. We do not pursue here this "finite" side of the theory further, which is devoted to the study of FCov(M) of finite coverings of the modular orbifold, and we turn our attention to the study of some special infinite covers.…”

Combinatorial categories and permutation groups

“…Similarly, Grothendieck's dessins d'enfants [21,22] are the finite coverings of a sphere minus three points, so their parent group is its fundamental group Γ = F 2 , with generators X, Y and Z inducing the monodromy permutations at the three punctures.…”

Fields of definition of uniform dessins on quasiplatonic surfaces

“…is one of the (2, 3, 7) uniform Belyi functions (see Example 4.44 in [9]) and it is defined over M (K) = Q. However, in this model there are automorphisms not defined over Q, and therefore we cannot expect all uniform Belyi functions in K F to be defined over Q simultaneously.…”