  For each d we construct CAT(0) cube complexes on which Cremona groups rank d
act by isometries. From these actions we deduce new and old group theoretical
and dynamical results about Cremona groups. In particular, we study the
dynamical behaviour of the irreducible components of exceptional loci, we prove
regularization theorems, we find new constraints on the degree growth for
non-regularizable birational transformations, and we show that the centralizer
of certain birational transformations is small.
