Article
Quasi-platonic PSL(2,q)-actions on Closed Riemann Surfaces
Albanian Journal of Mathematics
(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 pin 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 PSL(2,q) and the action of the absolute Galois group on PSL(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.
Keywords
- Riemann surface,
- quasi-platonic surface,
- automorphism group,
- symmetries
Disciplines
Publication Date
December, 2015
DOI
http://albanian-j-math.com/archives/2015-02.pdf
Citation Information
Sean A Broughton. "Quasi-platonic PSL(2,q)-actions on Closed Riemann Surfaces" Albanian Journal of Mathematics Vol. 9 Iss. 1 (2015) p. 31 - 61 ISSN: 1930-1235 Available at: http://works.bepress.com/allen_broughton/36/