#### Document Type

Article

#### Publication Date

12-2-2015

#### 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 *PSL _{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->S/G*is branched over

*{0, 1,infinity}*. 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

*S*by dihedrally symmetric polygons, generalizing the idea of a platonic solid. Each automorphism

*p*in the absolute Galois group determines a transform

*S*by transforming the coefficients of the defining equations of

^{p}*S*. The transform defines a possibly new quasi-platonic action and a transformed dessin

*D*. Here, in this paper, we describe the quasi-platonic actions of

^{p}*PSL*and the action of the absolute Galois group on

_{2}(q)*PSL*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.

_{2}(q)#### Recommended Citation

Broughton, Sean A., "Quasi-platonic PSL2(q)-actions on closed Riemann surfaces" (2015). *Mathematical Sciences Technical Reports (MSTR)*. 151.

http://scholar.rose-hulman.edu/math_mstr/151

## Comments

MSTR 15-01