Normal subgroups in the Cremona group

with an appendix by Yves de CornulierInternational audienceLet k be an algebraically closed field. We show that the Cremona group of all birational transformations of the projective plane P^2 over k 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 which generate non trivial normal subgroups