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Quasi-platonic PSL(2,q)-actions on Closed Riemann Surfaces
Albanian Journal of Mathematics (2015)
  • Sean A Broughton
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.
  • Riemann surface,
  • quasi-platonic surface,
  • automorphism group,
  • symmetries
Publication Date
December, 2015
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
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