The derived length of subgroups of the Cremona group

RESUMO

We prove that the maximal derived length of a solvable subgroup of the plane Cremona group over $\mathbb C$ is equal to $5$. We also show that the maximal derived length of a finite solvable subgroup is $4$, and give a short proof of the known sharp bound $5$ for bounded solvable subgroups. A major ingredient is a centraliser theorem: the centraliser of a finite solvable subgroup of derived length $4$ contains no loxodromic element. The proof relies on Blanc's classification of maximal algebraic subgroups and then uses equivariant Sarkisov theory, orbit estimates, superrigidity, and fixed curves of genus at least two.