Jordan bounds for volume-preserving Cremona groups

Abstract

We show that the Jordan constant for the volume-preserving plane Cremona group $\mathrm{Bir}(\mathbb P^2, \Delta)$ is $12$. We provide a Jordan bound of $144$ for the three-dimensional volume-preserving Cremona group $\mathrm{Bir}(\mathbb P^3,\Delta)$. We also provide a weak geometric Jordan bound of $2^{11} \cdot 3^2$ for $\mathrm{Bir}(\mathbb P^2)$.