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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 $\cal D$ 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 $\Gamma$, whose periods are determined by the valencies of the graph, and a finite index subgroup $H$ of $\Gamma$, uniquely determined up to conjugacy, such that the quotient of the upper half plane by $H$ is a Riemann surface $R({\cal D})$. 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 $\Gamma$. If moreover the dessin happens to have no free edges (i.e. it is clean), then $R({\cal D})$ becomes platonic. One first result in this paper asserts that quasiplatonic surfaces of genus $\leq 2$ 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 $\Gamma$ is an arithmetic Fuchsian group). The main focus of this paper is the study of the injectivity of the map ${\cal D}\mapsto R({\cal D})$. 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 $\Gamma$ in this case is of the form $(l,m,n)$. If $m\leq 2$ then the dessin becomes {a map}. The main theorem of this paper now states that if ${\cal D}_1$ and ${\cal D}_2$ are two non-arithmetic maps of maximal type, then $R({\cal D}_1)$ and $R({\cal D}_2)$ are conformally equivalent if and only if ${\cal D}_1$ and ${\cal D}_2$ are isomorphic maps. The maximal type of a uniform map translates into a condition on $l$ and $n$ for which the associated triangle group $\Gamma$ 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 [{\it 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
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