Actions of Cremona groups on CAT(0) cube complexes

TL;DR

This paper constructs CAT(0) cube complexes for Cremona groups of various ranks, enabling new insights into their algebraic and dynamical properties, including degree growth constraints and centralizer size.

Contribution

It introduces a novel geometric framework using CAT(0) cube complexes to analyze Cremona groups, revealing new group-theoretic and dynamical results.

Findings

Constructed CAT(0) cube complexes for Cremona groups of rank d

Established new constraints on degree growth of birational transformations

Proved smallness of centralizers for certain transformations

Abstract

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.