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Normal subgroups in the Cremona group

Acta Mathematica volume 210, pages 31–94 (2013)Cite this article

Abstract

Let k be an algebraically closed field. We show that the Cremona group of all birational transformations of the projective plane \( \mathbb{P}_{\mathbf{k}}^2 \) is not a simple group. The strategy makes use of hyperbolic geometry, geometric group theory and algebraic geometry to produce elements in the Cremona group that generate non-trivial normal subgroups.

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Author notes

  1. Serge Cantat

    Present address: Département de Mathématiques et Appl., École Normale Supérieure, 45 rue d’Ulm, FR-75230, Paris Cedex 5, France

  2. Stéphane Lamy

    Present address: Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118 route de Narbonne, FR-31062, Toulouse Cedex 9, France

Authors and Affiliations

  1. Université de Rennes I, Campus de Beaulieu, Bâtiment 22-23, FR-35042, Rennes Cedex, France

    Serge Cantat

  2. Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK

    Stéphane Lamy

  3. Laboratoire de Mathématiques d’Orsay, CNRS & Université Paris-Sud 11, Bâtiment 425, FR-91405, Orsay, France

    Yves de Cornulier

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  1. Serge Cantat
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  2. Stéphane Lamy
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  3. Yves de Cornulier
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Correspondence to Serge Cantat.

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With an appendix by Yves de Cornulier.

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Cantat, S., Lamy, S. & de Cornulier, Y. Normal subgroups in the Cremona group. Acta Math 210, 31–94 (2013). https://doi.org/10.1007/s11511-013-0090-1

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  • DOI: https://doi.org/10.1007/s11511-013-0090-1