The Cremona group and the Picard–Manin space

The Cremona group is the group of birational transformations of the projective plane. One of the key techniques to understand its group structure is its isometric action on an infinite dimensional hyperboloid. We will explain this action and give some examples on how it can be used to prove some structure results about the Cremona group. In particular, we will show that the Cremona group is not simple and describe its simple subgroups.