Normal subgroups in the Cremona group

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.