no. 3
The real plane Cremona group is an amalgamated product
Zimmermann, Susanna1
1 Univ Angers, CNRS, LAREMA, SFR MATHSTIC, F-49000 Angers, France
Annales de l'Institut Fourier, Volume 71 (2021) no. 3, pp. 1023-1045.

We show that the real Cremona group of the plane is a non-trivial amalgam of two groups amalgamated along their intersection and give an alternative proof of its abelianisation.

On montre que le groupe de Cremona du plan réel est un produit amalgamé non-trivial de deux groupes le long de leur intersection et on donne une preuve alternative de son abélianisation.

Received:
Revised:
Accepted:
Online First:
Published online:
DOI: 10.5802/aif.3415
Classification: 14E07, 20F05, 14P99
Keywords: Cremona groups, amalgamated product
Mot clés : Groupe de Cremona, produit amalgamé

Zimmermann, Susanna 1

1 Univ Angers, CNRS, LAREMA, SFR MATHSTIC, F-49000 Angers, France
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Zimmermann, Susanna. The real plane Cremona group is an amalgamated product. Annales de l'Institut Fourier, Volume 71 (2021) no. 3, pp. 1023-1045. doi : 10.5802/aif.3415. https://aif.centre-mersenne.org/articles/10.5802/aif.3415/

[1] Blanc, Jérémy Simple relations in the Cremona group, Proc. Am. Math. Soc., Volume 140 (2012) no. 5, pp. 1495-1500 | DOI | MR | Zbl

[2] Blanc, Jérémy; Lamy, Stéphane; Zimmermann, Susanna Quotients of higher dimensional Cremona groups (2019) to be published in Acta Mathematica, volume 226 (2021), issue 2, https://arxiv.org/abs/1901.04145

[3] Blanc, Jérémy; Mangolte, Frédéric Cremona groups of real surfaces, Automorphisms in birational and affine geometry (Springer Proceedings in Mathematics & Statistics), Volume 79, Springer, 2014, pp. 35-58 (Papers based on the presentations at the conference, Levico Terme, Italy, October 29 – November 3, 2012) | MR | Zbl

[4] Cantat, Serge; Lamy, Stéphane; de Cornulier, Yves Normal subgroups in the Cremona group, Acta Math., Volume 210 (2013) no. 1, pp. 31-94 (With an appendix by Yves de Cornulier) | DOI | MR | Zbl

[5] Castelnuovo, Guido Le trasformazioni generatrici del gruppo cremoniano nel piano, Torino Atti (1901) no. 36, pp. 861-874 | Zbl

[6] Corti, Alessio Factoring birational maps of threefolds after Sarkisov, J. Algebr. Geom., Volume 4 (1995) no. 2, pp. 223-254 | MR | Zbl

[7] Hacon, Christopher D.; McKernan, James The Sarkisov program, J. Algebr. Geom., Volume 22 (2013) no. 2, pp. 389-405 | DOI | MR | Zbl

[8] Iskovskikh, Vasiliĭ Alekseevich Proof of a theorem on relations in the two-dimensional Cremona group, Usp. Mat. Nauk, Volume 40 (1985) no. 5(245), pp. 255-256 | MR | Zbl

[9] Iskovskikh, Vasiliĭ Alekseevich Factorization of birational mappings of rational surfaces from the point of view of Mori theory, Usp. Mat. Nauk, Volume 51 (1996) no. 4(310), pp. 3-72 | MR | Zbl

[10] Lamy, St’ephane Groupes de transformations birationnelles de surfaces, 2010 (Mémoire d’habilitation à diriger des recherches, Université Claude Bernard Lyon 1, France)

[11] Lamy, St’ephane; Zimmermann, Susanna Signature morphisms from the Cremona group over a non-closed field, J. Eur. Math. Soc., Volume 22 (2020) no. 10, pp. 3133-3173 | DOI | MR | Zbl

[12] Robayo, Maria Fernanda; Zimmermann, Susanna Infinite algebraic subgroups of the real Cremona group, Osaka J. Math., Volume 55 (2018) no. 4, pp. 681-712 | MR | Zbl

[13] Ronga, Felice; Vust, Thierry Birational diffeomorphisms of the real projective plane, Comment. Math. Helv., Volume 80 (2005) no. 3, pp. 517-540 | MR | Zbl

[14] Sansuc, Jean-Jacques Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres, J. Reine Angew. Math., Volume 327 (1981), pp. 12-80 | Zbl

[15] Serre, Jean-Pierre Trees, Springer Monographs in Mathematics, Springer, 2003 (Translated from the French original by John Stillwell) | Zbl

[16] Wright, David Two-dimensional Cremona groups acting on simplicial complexes, Trans. Am. Math. Soc., Volume 331 (1992) no. 1, pp. 281-300 | DOI | MR | Zbl

[17] Yasinsky, Egor Subgroups of odd order in the real plane Cremona group, J. Algebra, Volume 461 (2016), pp. 87-120 | DOI | MR | Zbl

[18] Zimmermann, Susanna The abelianisation of the real Cremona group, Duke Math. J., Volume 167 (2018) no. 2, pp. 211-267 | MR | Zbl

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