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Arbitrarily large Galois orbits of non-homeomorphic surfaces

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Abstract

It has recently been proved that any element \(\sigma \) of the absolute Galois group \(\mathrm{Gal}\,\overline{\mathbb {Q}}/{\mathbb Q}\) different from the identity and complex conjugation conjugates some surface S into a surface \(S^{\sigma }\) with non-isomorphic fundamental group. Here we use group theory to construct, for each integer \(n\geqslant 0\), an n-dimensional family of complex surfaces whose conjugates under \(\mathrm{Gal}\,\overline{\mathbb {Q}}/{\mathbb Q}\) exhibit arbitrarily many non-isomorphic fundamental groups. These fundamental groups nevertheless have isomorphic profinite completions. The surfaces constructed are isogenous to higher products via actions of groups \(\mathrm{PGL}_2(p)\) on algebraic curves.

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Authors and Affiliations

  1. Departamento de Matemáticas, Universidad Autónoma de Madrid, Ciudad Universitaria de Cantoblanco, 28049, Madrid, Spain

    Gabino González-Diez

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

    Gareth A. Jones

  3. Fachrichtung Mathematik, Universität des Saarlandes, 66123, Saarbrücken, Germany

    David Torres-Teigell

Authors
  1. Gabino González-Diez
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  2. Gareth A. Jones
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  3. David Torres-Teigell
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Correspondence to Gabino González-Diez.

Additional information

The second author is grateful to the Departamento de Matemáticas, Universidad Autónoma de Madrid, for financially supporting a visit during which much of this research was carried out. The first and third authors are partially supported by the Grant MTM2012-31973 of the MICINN. The first two authors thank the ICMS in Edinburgh, where they started collaborating on Beauville surfaces at a September 2008 workshop on the Grothendieck–Teichmüller Theory of Dessins d’Enfants.

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González-Diez, G., Jones, G.A. & Torres-Teigell, D. Arbitrarily large Galois orbits of non-homeomorphic surfaces. European Journal of Mathematics 4, 223–241 (2018). https://doi.org/10.1007/s40879-017-0203-z

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  • DOI: https://doi.org/10.1007/s40879-017-0203-z

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