The Abelianization of the real Cremona group

Abstract

We present the abelianisation of the group of birational transformations of P 2 R .

The real plane Cremona group is an amalgamated product

“…For the finite subgroups of odd order Corollary 1.2 can also be verified by checking their classification in [17]. An earlier version of this article used an explicit presentation of Bir R (P 2 ) and of G • in terms of generators and generating relations, the first of which is proven in [18] and the second was proven analogously in the earlier version of this article. The present version does not use either presentation.…”

“…We now reprove [18,Proposition 5.3] by using the principle idea of [11] in the construction of a homomorphism Bir(P 2 k ) −→˚I Z/2Z over a perfect field k. Proof. -We denote by BirMori(P 2 ) the set of birational transformations between rank 1 fibrations.…”

Generating the plane Cremona groups by involutions

“…We mention that over the field of real numbers, the generation of Bir R (P 2 ) by involutions was proved by S. Zimmermann in her PhD thesis [Zim16,Corollary II.4.12], together with a complete description of the abelianization of Bir R (P 2 ) [Zim18]. Over the field with 2 elements, the generation of Bir F 2 (P 2 ) by involutions was established by the second author [Sch20], with a similar strategy as in the present paper, but with some cases relying on an exhaustive search assisted by computer.…”

Quotients of groups of birational transformations of cubic del Pezzo fibrations

“…Later this result was generalized by A. Lonjou to an arbitrary base field [Lon16]. Another approach to non-simplicity of Bir R (P 2 ), based on explicit presentation of Bir R (P 2 ) by generators and relations, was discovered by S. Zimmermann in [Zim18]. The question of the non-simplicity of higher rank Cremona groups remained open until the recent preprint [BLZ19], where the so-called Sarkisov program and recent achievements in the Borisov-Alexeev-Borisov conjecture [Bir19] allowed the authors to prove the following result:…”