QUASI-PLATONIC PSL2(q)-ACTIONS ON CLOSED RIEMANN SURFACES

Abstract

This paper is the first of two papers whose combined goal is to explore the dessins d’enfant and symmetries of quasi-platonic actions of $PS{L}_2(q)$. A quasi-platonic action of a group $G$ on a closed Riemann $S$ surface is a conformal action for which $S/G$ is a sphere and $S\to S/G$ is branched over $\{0,1,\infty \}$. The unit interval in $S/G$ may be lifted to a dessin d’enfant $\mathcal{D}$, an embedded bipartite graph in $S$. The dessin forms the edges and vertices of a tiling on $S$ by dihedrally symmetric polygons, generalizing the idea of a platonic solid. Each automorphism $\psi$ in the absolute Galois group determines a transform $S^{\psi }$ by transforming the coefficients of the defining equations of $S$. The transform defines a possibly new quasi-platonic action and a transformed dessin ${\mathcal{D}}^{\psi }$.

Here, in this paper, we describe the quasi-platonic actions of $PS{L}_2(q)$ and the action of the absolute Galois group on $PS{L}_2(q)$ actions. The second paper discusses the quasi-platonic actions constructed from symmetries (reflections) and the resulting triangular tiling that refines the dessin d’enfant. In particular, the number of ovals and the separation properties of the mirrors of a symmetry are determined.