Normal subgroups in the Cremona group

@article{Cantat2010NormalSI,
  title={Normal subgroups in the Cremona group},
  author={Serge Cantat and St'ephane Lamy and Yves Cornulier},
  journal={Acta Mathematica},
  year={2010},
  volume={210},
  pages={31-94}
}
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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