For each d, we construct CAT(0) cube complexes on which Cremona groups of rank d act by isometries. From these actions deduce new and old group-theoretical dynamical results about groups. In particular, study the behavior irreducible components exceptional loci. This leads to proofs regularization theorems, such as with property FW. We also find constraints degree growth for non-pseudo-regulari...

We construct families of $CAT(0)$ groups containing finitely presented whose divergence functions are the form $r^\alpha$ for a dense set exponents $\alpha \in [2,\infty)$ and $r^q\log(r)$ integers $q \geq 2$. The same construction also yields examples contracting elements which not in certain subgroups.