Fuchsian triangle groups and Grothendieck dessins. Variations on a theme of Belyi
According to a theorem of Belyi, a smooth projective algebraic curve \(X\) is defined over a number field if and only if there is a non-constant element of its function field ramified only over 0, 1, and \(\infty\). The existence of such a Belyi function \(\beta\) is equivalent to that of a representation of the curve as a possibly compactified quotient of the Poincaré upper half plane by a subgroup of finite index in a Fuchsian triangle group. If in particular \(\beta\) ramifies in all points of \(\beta^{-1}(1)\) with ramification order 2, the graph \(\beta^{-1} ([0,1])\) on \(X\) is called a ``Grothendieck dessin'', and it is shown that \(X\) is uniquely determined up to isomorphisms already by the topology of its Grothendieck dessins. Examples such as the Platonic solids on the sphere and natural triangulations of the Fermat curves are discussed in detail, the interplay with combinatorial objects as the ``cartographic group'' of the dessin is described, and connections with Shimura varieties and parts of physics like rational triangular billards are indicated.
scientific article; zbMATH DE number 617384
scientific article