Centralizers of elements of infinite order in plane Cremona groups

Abstract

Let $\mathbf{K}$ be an algebraically closed field. The Cremona group ${\mathrm{Cr}}_2(\mathbf{K})$ is the group of birational transformations of the projective plane ${\mathbb{P}}_{\mathbf{K}}^2$. We carry out an overall study of centralizers of elements of infinite order in ${\mathrm{Cr}}_2(\mathbf{K})$ which leads to a classification of embeddings of ${\mathbf{Z}}^2$ into ${\mathrm{Cr}}_2(\mathbf{K})$, as well as a classification of maximal non-torsion abelian subgroups of ${\mathrm{Cr}}_2(\mathbf{K})$ when $\mathrm{char}(\mathbf{K})=0$.