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

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 PSL2(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 → S/G is branched over {0, 1, ∞}. The unit interval in S/G may be lifted to a dessin d'enfant D, an embedded bipartite graph in S. The dessin forms the edges and vertices of a tiling on Sψ by dihedrally symmetric polygons, generalizing the idea of a platonic solid. Each automorphism ψ in the absolute Galois group determines a transform S by transforming the coefficients of the defining equations of S. The transform defines a possibly new quasi-platonic action and a transformed dessin Dψ. Here, in this paper, we describe the quasi-platonic actions of PSL2(q) and the action of the absolute Galois group on PSL2(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. [ABSTRACT FROM AUTHOR]