Jordan bounds for volume-preserving Cremona groups

Abstract

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

20 pages, 3 figures