Work

The derived length of subgroups of the Cremona group

Year: 2026

Type: preprint

Abstract: We prove that the maximal derived length of a solvable subgroup of the plane Cremona group over $\mathbb C$ is equal to $5$. The main difficulty is the case of subgroups containing loxodromic elements. To treat this case, we prove that the maximal derived length of a finite solvable subgroup is $4$ and establish a centraliser theorem: the centraliser of a finite solvable subgroup of derived length $4$ contains no loxodromic element. The proof of the centraliser theorem 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.

Source: arXiv (Cornell University)

DOI: 10.48550/arxiv.2609.08876

Authors: Furter, Jean-Philippe, Hedén, Isac

Institutions: Uppsala University, Centre National de la Recherche Scientifique, Université de Bordeaux, Institut de Mathématiques de Bordeaux

Language: English

Topic: Algebraic Geometry and Number Theory

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

OpenAlex ID: W7212021746

Cites: 2

Cited by:

Related to: 10