Cyclic Projective Planes and Wada Dessins

Bipartite graphs occur in many parts of mathematics, and their embeddings into orientable compact surfaces are an old subject. A new interest comes from the fact that these embeddings give {\em dessins d'enfants} providing the surface with a unique structure as a Riemann surface and algebraic curve. In this paper, we study the (surprisingly many different) dessins coming from the graphs of finite cyclic projective planes. It turns out that all reasonable questions about these dessins --- uniformity, regularity, automorphism groups, cartographic groups, defining equations of the algebraic curves, their fields of definition, Galois actions --- depend on {\em cyclic orderings} of difference sets for the projective planes. We explain the interplay between number theoretic problems concerning these cyclic ordered difference sets and topological properties of the dessin like e.g. the {\em Wada property} that every vertex lies on the border of every cell.

2000 Mathematics Subject Classification: 51E15, 05C10, 14H25, 14H55, 20H10, 30F10

Keywords and Phrases: Projective planes, difference sets, dessins d'enfants, Riemann surfaces, Fuchsian groups, algebraic curves

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