On solvable subgroups of the Cremona group

Abstract

The Cremona group $\operatorname{Bir}(\mathbb{P}^{2}_{\mathbb{C}})$ is the group of birational self-maps of $\mathbb{P}^{2}_{\mathbb{C}}$. Using the action of $\operatorname{Bir}(\mathbb{P}^{2}_{\mathbb{C}})$ on the Picard-Manin space of $\mathbb{P}^{2}_{\mathbb{C}}$, we characterize its solvable subgroups. If $\mathrm{G}\subset\operatorname{Bir}(\mathbb{P}^{2}_{\mathbb{C}})$ is solvable, nonvirtually Abelian, and infinite, then up to finite index: either any element of $\mathrm{G}$ is of finite order or conjugate to an automorphism of $\mathbb{P}^{2}_{\mathbb{C}}$, or $\mathrm{G}$ preserves a unique fibration that is rational or elliptic, or $\mathrm{G}$ is, up to conjugacy, a subgroup of the group generated by one hyperbolic monomial map and the diagonal automorphisms.

We also give some corollaries.