Subgroups of elliptic elements of the Cremona group

TL;DR

This paper classifies elliptic-element subgroups of the Cremona group, describes their structure, and proves the Tits alternative for its subgroups, advancing understanding of its algebraic properties.

Contribution

It provides a classification of elliptic-element subgroups and establishes the Tits alternative for all subgroups of the Cremona group.

Findings

Classification of elliptic-element subgroups

Proof of the Tits alternative for Cremona subgroups

Description of solvable subgroups and their derived length

Abstract

The Cremona group is the group of birational transformations of the complex projective plane. In this paper we classify its subgroups that consist only of elliptic elements using elementary model theory. This yields in particular a description of the structure of torsion subgroups. As an appliction, we prove the Tits alternative for arbitrary subgroups of the Cremona group, generalizing a result of Cantat. We also describe solvable subgroups of the Cremona group and their derived length, refining results from D\'eserti.