Centralizers of elements of infinite order in plane Cremona groups

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