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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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