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 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 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 Wada property that every vertex lies on the border of every cell.

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Metadaten
Author:Manfred Streit, Jürgen WolfartGND
URN:urn:nbn:de:hebis:30:3-451491
URL:https://www.math.uni-bielefeld.de/documenta/vol-06/04.pdf
ISSN:1431-0635
Parent Title (German):Documenta mathematica
Publisher:Deutsche Mathematiker-Vereinigung
Place of publication:Bielefeld
Document Type:Article
Language:English
Year of Completion:2001
Year of first Publication:2001
Publishing Institution:Universitätsbibliothek Johann Christian Senckenberg
Release Date:2017/11/27
Tag:Fuchsian groups; Riemann surfaces; algebraic curves; dessins d’enfants; difference sets; projective planes
Volume:6
Page Number:30
First Page:39
Last Page:68
HeBIS-PPN:425118150
Institutes:Informatik und Mathematik / Mathematik
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
MSC-Classification:05-XX COMBINATORICS (For finite fields, see 11Txx) / 05Cxx Graph theory (For applications of graphs, see 68R10, 81Q30, 81T15, 82B20, 82C20, 90C35, 92E10, 94C15) / 05C10 Planar graphs; geometric and topological aspects of graph theory [See also 57M15, 57M25]
14-XX ALGEBRAIC GEOMETRY / 14Hxx Curves / 14H25 Arithmetic ground fields [See also 11Dxx, 11G05, 14Gxx]
14-XX ALGEBRAIC GEOMETRY / 14Hxx Curves / 14H55 Riemann surfaces; Weierstrass points; gap sequences [See also 30Fxx]
20-XX GROUP THEORY AND GENERALIZATIONS / 20Hxx Other groups of matrices [See also 15A30] / 20H10 Fuchsian groups and their generalizations [See also 11F06, 22E40, 30F35, 32Nxx]
30-XX FUNCTIONS OF A COMPLEX VARIABLE (For analysis on manifolds, see 58-XX) / 30Fxx Riemann surfaces / 30F10 Compact Riemann surfaces and uniformization [See also 14H15, 32G15]
51-XX GEOMETRY (For algebraic geometry, see 14-XX) / 51Exx Finite geometry and special incidence structures / 51E15 Affine and projective planes
Sammlungen:Universitätspublikationen
Licence (German):License LogoDeutsches Urheberrecht

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