Sur les automorphismes du groupe de Cremona

Abstract

We prove that an automorphism of the group of birational transformations of the complex projective plane is the composition of an interior automorphism and an automorphism of the field C. The proof is based on a study of maximal abelian subgroups of the Cremona group.

Normal subgroups in the Cremona group

“…Blanc proved in [3] that Bir(P 2 k ) has only two closed normal subgroups for this topology, namely {Id} and Bir(P 2 k ) itself. Déserti proved that Bir(P 2 C ) is perfect, hopfian, and co-hopfian, and that its automorphism group is generated by inner automorphisms and the action of automorphisms of the field of complex numbers (see [17,18,19]). In particular, there is no obvious algebraic reason which explains why Bir(P 2 C ) is not simple.…”

Birational maps preserving the contact structure on $\mathbb{P}^3_\mathbb{C}$

“…Even if Bir(P 2 ) has not the same structure as Aut(C 2 ) (see Appendix of [11]) the automorphisms group of Bir(P 2 ) can be described and a similar result is obtained ( [17]). We now would like to describe the group Aut…”

On solvable subgroups of the Cremona group

“…• φ is contained in a maximal abelian subgroup denoted Ab(φ) that preserves y = cst fiberwise ( [13]), • the centralizer of φ is a finite extension of Ab(φ) (see [9,Theorem B]).…”

“…We know properties on finite subgroups ( [16]), finitely generated subgroups ( [6]), uncountable maximal abelian subgroups ( [13]), nilpotent subgroups ( [14]) of the Cremona group. In this article we focus on solvable subgroups of the Cremona group.…”