Subgroups of elliptic elements of the Cremona group

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 application, 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 Déserti.

Birational Kleinian groups

“…Lamy [Lam01] proved that the group Aut(C 2 ) of polynomial automorphisms of the affine plane C 2 satisfies the Tits alternative. For Bir(P 2 ), the Tits alternative has been proved by Cantat [Can11] for finitely generated subgroups and in general by Urech [Ure21]. Furthermore for Bir(P 2 ) the Tits alternative has been improved to the following classification of infinite subgroups; one proves the Tits alternative by proving it for each case in the classification.…”

“…Furthermore for Bir(P 2 ) the Tits alternative has been improved to the following classification of infinite subgroups; one proves the Tits alternative by proving it for each case in the classification. The bulk of the classification is done by Cantat [Can11] and it is finished by Urech [Ure21]. Besides [Can11] and [Ure21], the works of Weil [Wei55], Gizatullin [Giz80], Blanc-Cantat [BC16] and Déserti [Dés15] are respectively the main input for case 1), 4), 5), 6) in the classification.…”

“…The bulk of the classification is done by Cantat [Can11] and it is finished by Urech [Ure21]. Besides [Can11] and [Ure21], the works of Weil [Wei55], Gizatullin [Giz80], Blanc-Cantat [BC16] and Déserti [Dés15] are respectively the main input for case 1), 4), 5), 6) in the classification.…”

Centralizers of elements of infinite order in plane Cremona groups

“…Thus, in order to obtain a classification of embeddings of Z 2 in Cr 2 (K), we need a detailed study of centralizers of base-wandering Jonquières twists, which is the main task of this article. Regarding the elements of finite order and their centralizers in Cr 2 (K), the problem is of a rather different flavour and we refer the readers to [Bla07], [DI09], [Ser10], [Ure18] and the references therein.…”