Graphs on surfaces and algebraic curves

A graph, embedded into a compact oriented surface, defines (under certain conditions) a complex structure on this surface; sometimes the additional real parameters define families of complex structures. Several constructions of this kind are known; the simplest one is based on the metrized triangulations and serves as a discretization of the procedure of defining complex structure by a riemannian metric on a surface.
It will be shown in the talk, that all these constructions are covered by the Grothendieck's theory of dessins d'enfants and its generalizations. In the frames of this theory the curves over the algebraic curves over number fields are naturally distinguished; the absolute Galois group in $\mathrm{Aut}(\overline{\mathbb Q})$ arises as a group of “hidden” symmetries. Some examples of the correspondences between the combinatorial-topological and algebro-geometrical structures will be given — on the level of individual curves as well as from the viewpoint of the geometry of moduli spaces $\mathcal M_g(\overline{\mathbb Q})\subset\mathcal M_g(\mathbb C)$ of all the curves of a given genus. The relations of the dessins d'enfants theory with several domains of mathematics and physics will be mentioned.