# Centralizers of elements of infinite order in plane Cremona groups

## TL;DR

This paper studies the structure of centralizers of infinite order elements in the plane Cremona group, classifying certain abelian subgroups and embeddings, especially over fields of characteristic zero.

## Contribution

It provides a classification of embeddings of  into the Cremona group and describes maximal non-torsion abelian subgroups, advancing understanding of the group's structure.

## Key findings

- Classification of  embeddings into (Cr_2(K))
- Description of maximal non-torsion abelian subgroups
- Analysis of centralizers of infinite order elements

## Abstract

Let $\mathbf{K}$ be an algebraically closed field. The Cremona group $\operatorname{Cr}_2(\mathbf{K})$ is the group of birational transformations of the projective plane $\mathbb{P}^2_{\mathbf{K}}$. 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 $\mathbf{Z}^2$ into $\operatorname{Cr}_2(\mathbf{K})$, as well as a classification of maximal non-torsion abelian subgroups of $\operatorname{Cr}_2(\mathbf{K})$ when $\operatorname{char}(\mathbf{K})=0$.