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The Riemann surface of a uniform dessin. (English) Zbl 1064.14030

This paper looks into the relationship between dessins d’enfants and their associated Riemann surfaces. A dessin 𝒟 on a topological surface is an embedding of a bipartite graph whose complement has simply connected components. Associated to such a dessin there is a triangle group Γ, whose periods are determined by the valencies of the graph, and a finite index subgroup H of Γ, uniquely determined up to conjugacy, such that the quotient of the upper half plane by H is a Riemann surface R(𝒟). This is referred to as the Riemann surface underlying the dessin. The surface is smooth Belyi if H is torsion-free. The surface is quasiplatonic if the dessin is regular, meaning in addition that H is normal in Γ. If moreover the dessin happens to have no free edges (i.e. it is clean), then R(𝒟) becomes platonic. One first result in this paper asserts that quasiplatonic surfaces of genus 2 are platonic, and then gives an example in genus 3 where this is no longer true. The authors also observe that every smooth dessin of genus 2 is arithmetic (i.e. the associated triangle group Γ is an arithmetic Fuchsian group).

The main focus of this paper is the study of the injectivity of the map 𝒟R(𝒟). The authors specialize to the family of dessins which are uniform maps and hence ask whether two non-isomorphic such dessins can have the same underlying Riemann surface. A dessin is uniform if all black vertices (in the bipartite description) have equal valency l, all white vertices have equal valency m and all cells have equal valency 2n. The associated triangle group Γ in this case is of the form (l,m,n). If m2 then the dessin becomes a map. The main theorem of this paper now states that if 𝒟 1 and 𝒟 2 are two non-arithmetic maps of maximal type, then R(𝒟 1 ) and R(𝒟 2 ) are conformally equivalent if and only if 𝒟 1 and 𝒟 2 are isomorphic maps. The maximal type of a uniform map translates into a condition on l and n for which the associated triangle group Γ is not contained in a larger triangle group. The proof of the main theorem is based on a fundamental theorem of Margulis on commensurators of Fuchsian groups [R. J. Zimmer, Ergodic theory and semisimple groups (1984; Zbl 0571.58015)].

The paper ends with a list of examples where distinct maps and dessins may lie on the same Riemann surface, and also gives a comparison between the automorphism group of a dessin and the one for its underlying Riemann surface.

14H55Riemann surfaces; Weierstrass points; gap sequences
30F10Compact Riemann surfaces; uniformization