How symmetric can maps on surfaces be?

Širáň, Jozef (2013). How symmetric can maps on surfaces be? In: Blackburn, Simon R.; Gerke, Stefanie and Wildon, Mark eds. Surveys in Combinatorics 2013. London Mathematical Society Lecture Note Series. Cambridge: Cambridge University Press, pp. 161–238.


a map, that is, a cellular embeding of a gaph on a surface, may admit symmetries such as rotations and reflections. Prominent examples of maps with a 'high level of symmetry' come from Platonic and Archimedean solids. The theory of maps and their symmetries is surprisingly rich and interacts with other disciplines in mathematics such as algebraic topology, group theory, hyperbolic geometry, the theory of Riemann surfaces and Galois theory.
In the first half of the paper we outline the fundamentals of the algebraic theory of regular and orientably regular maps. The second half of the article is a survey of the state-of-the-art with respect to the classification of such maps by their automorphism groups, underlying graphs, and supporting surfaces. We conclude by introducing the notion of 'external symmetries' of regular maps, going well byeyond automorphisms, and discuss the corresponding 'super-symmetric' maps.

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  • Item ORO ID
  • 41750
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  • Book Section
  • ISBN
  • 1-107-65195-6, 978-1-107-65195-1
  • Extra Information
  • This volume contains nine survey articles based on the invited lectures given at the 24th British Combinatorial Conference, held at Royal Holloway, University of London in July 2013. This biennial conference is a well-established international event, with speakers from around the world. The volume provides an up-to-date overview of current research in several areas of combinatorics, including graph theory, matroid theory and automatic counting, as well as connections to coding theory and Bent functions. Each article is clearly written and assumes little prior knowledge on the part of the reader. The authors are some of the world's foremost researchers in their fields, and here they summarise existing results and give a unique preview of cutting-edge developments. The book provides a valuable survey of the present state of knowledge in combinatorics, and will be useful to researchers and advanced graduate students, primarily in mathematics but also in computer science and statistics.
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  • Faculty of Science, Technology, Engineering and Mathematics (STEM) > Mathematics and Statistics
    Faculty of Science, Technology, Engineering and Mathematics (STEM)
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  • © 2013 Cambridge University Press
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  • Jozef Širáň