Actions of Cremona groups on CAT(0) cube complexes

Abstract

For each d, we construct $\mathrm{CAT}(0)$ cube complexes on which Cremona groups of 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 behavior of the irreducible components of exceptional loci. This leads to proofs of regularization theorems, such as the regularization of groups with property FW. We also find new constraints on the degree growth for non-pseudo-regularizable birational transformations, and we show that the centralizer of certain birational transformations is small.