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.

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.

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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 how many different regular dessins of the same type can exist on a given quasiplatonic Riemann surface.

We show that the action of the absolute Galois group on dessins d'enfants of given genus g is faithful, a result that had been previously established for g = 0 and g = 1.