I shall discuss the action of the Cremona group on the infinite dimensional space of b-divisors (where b stands for birational)\, obtained as a limit of all N�ron-Severi spaces for all possible blowups dominating the projective plane. This space is naturally endowed with a quadratic form of Minkowski type\, and so contains an infinite dimensional hyperboloid. One application of this construction is to produce many normal subgroups in the Cremona group\, via a general result in small cancellation theory. More generally\, the action allows to classify elements or subgroups or the Cremona group along the familiar elliptic/parabolic/loxodromic subtypes in hyperbolic geometry. I plan to discuss a few classification results along these lines\, for instance subgroups that contain a positive dimensional algebraic normal subgroup\, or subgroups of elliptic elements that do not preserve a pencil of rational curves.