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Beauville Surfaces and Groups: A Survey

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Rigidity and Symmetry

Part of the book series: Fields Institute Communications ((FIC,volume 70))

Abstract

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?

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Notes

  1. 1.

    Here, as is customary in algebraic geometry, a ‘surface’ is an algebraic variety which is 2-dimensional over the field of coefficients; in this case, that field is \(\mathbb{C}\) so these surfaces have dimension 4 as real manifolds. Rather confusingly, a complex algebraic curve, 1-dimensional over \(\mathbb{C}\), can be regarded as a Riemann surface, where ‘surface’ now indicates 2-dimensionality over \(\mathbb{R}\)!

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Acknowledgements

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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  1. School of Mathematics, University of Southampton, Southampton, SO17 1BJ, UK

    Gareth A. Jones

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  1. Gareth A. Jones
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  1. Department of Mathematics, Cornell University, Ithaca, New York, USA

    Robert Connelly

  2. Department of Mathematics and Statistics, York University, Toronto, Ontario, Canada

    Asia Ivić Weiss

  3. Department of Mathematics & Statistics, York University, Toronto, Ontario, Canada

    Walter Whiteley

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Jones, G.A. (2014). Beauville Surfaces and Groups: A Survey. In: Connelly, R., Ivić Weiss, A., Whiteley, W. (eds) Rigidity and Symmetry. Fields Institute Communications, vol 70. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0781-6_11

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