Actions of Cremona groups on CAT(0) cube complexes

Understand

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.