In this expository talk, we will introduce the Picard-Manin space and the infinite-dimensional hyperbolic space associated with its intersection form. The group of birational transformations of the complex projective plane, called the Cremona group, acts naturally by isometries on this hyperbolic space. The talk will be motivated by Cantat and Lamy's proof that the Cremona group is not simple. This was one of the first instances of the use of an infinite-dimensional hyperbolic space in geometric group theory. I will first describe the Picard-Manin space along with the action of the Cremona group, then discuss how ideas coming from small cancellation theory lead to the construction of nontrivial proper normal subgroups.