On solvable subgroups of the Cremona group

TL;DR

This paper characterizes the structure of solvable subgroups within the Cremona group of birational maps of the complex projective plane, revealing their possible forms and properties.

Contribution

It provides a classification of infinite, non-abelian solvable subgroups of the Cremona group using geometric and algebraic tools.

Findings

Solvable subgroups are either finite order elements, automorphisms, preserve a fibration, or are generated by a hyperbolic monomial map and automorphisms.

Characterizes the possible algebraic structures of these subgroups.

Provides corollaries related to the structure and properties of these subgroups.

Abstract

The Cremona group $\mathrm{Bir}(\mathbb{P}^2_\mathbb{C})$ is the group of birational self-maps of $\mathbb{P}^2_\mathbb{C}$. Using the action of $\mathrm{Bir}(\mathbb{P}^2_\mathbb{C})$ on the Picard-Manin space of $\mathbb{P}^2_\mathbb{C}$ we characterize its solvable subgroups. If $\mathrm{G}\subset\mathrm{Bir}(\mathbb{P}^2_\mathbb{C})$ is solvable, non 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.