Almost simple subgroups of the space Cremona group

Source abstract

We prove that every finite almost simple subgroup that is not simple of the group of birational automorphisms of the three-dimensional complex projective space is isomorphic to one of the following: $\mathfrak{S}_5$, $\mathfrak{S}_6$, $\mathrm{PGL}_2(\mathbb{F}_7)$ or $\mathrm{PGL}_2(\mathbb{F}_9)$. We also prove that the group of birational automorphisms of the three-dimensional complex projective space contains unique (up to conjugation) subgroups isomorphic to $\mathrm{PGL}_2(\mathbb{F}_7)$ and $\mathrm{PGL}_2(\mathbb{F}_9)$.