Beauville Surfaces and Groups: A Survey

  • Gareth A. JonesEmail author
Part of the Fields Institute Communications book series (FIC, volume 70)


This is a survey of recent progress on Beauville surfaces, concentrating almost entirely on the group-theoretic and combinatorial problems associated with them. A Beauville surface \(\mathcal{S}\) is a complex surface formed from two orientably regular hypermaps of genus at least 2 (viewed as compact Riemann surfaces and hence as algebraic curves), with the same automorphism group G acting freely on their product. The following questions are discussed: Which groups G (called Beauville groups) have this property? What can be said about the automorphism group and the fundamental group of \(\mathcal{S}\)? Beauville surfaces are defined (as algebraic varieties) over the field \(\overline{\mathbb{Q}}\) of algebraic numbers, so how does the absolute Galois group \(\mathrm{Gal}\,\overline{\mathbb{Q}}/\mathbb{Q}\) act on them?


Beauville surface Beauville group Hypermap Compact Riemann surface Absolute Galois group 

Subject Classifications

Primary 20B25 Secondary 05C10 11G32 14J25 14J50 51M20 



The author is grateful to Gabino González-Diez and Bernhard Köck for some very helpful comments on an earlier draft of this paper, and to the organisers of the Workshop on Symmetry in Graphs, Maps and Polytopes at the Fields Institute, Toronto, 24–27 October 2011, for the opportunity to give a talk on which it is based.


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© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.School of MathematicsUniversity of SouthamptonSouthamptonUK

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