Quasi-platonic PSL2(q)-actions on closed Riemann surfaces

Authors

Sean A. Broughton, Rose-Hulman Institute of TechnologyFollow

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 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,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^p by transforming the coefficients of the defining equations of S. The transform defines a possibly new quasi-platonic action and a transformed dessin D^p. 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.

Comments

MSTR 15-01

Recommended Citation

Broughton, Sean A., "Quasi-platonic PSL2(q)-actions on closed Riemann surfaces" (2015). Mathematical Sciences Technical Reports (MSTR). 151.
https://scholar.rose-hulman.edu/math_mstr/151