no. 3
The real plane Cremona group is an amalgamated product
[Sur l’un des premiers problèmes de Wiles]
Zimmermann, Susanna1
1 Univ Angers, CNRS, LAREMA, SFR MATHSTIC, F-49000 Angers, France
Annales de l'Institut Fourier, Tome 71 (2021) no. 3, pp. 1023-1045.

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.

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.

Reçu le :
Révisé le :
Accepté le :
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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 
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Zimmermann, Susanna. The real plane Cremona group is an amalgamated product. Annales de l'Institut Fourier, Tome 71 (2021) no. 3, pp. 1023-1045. doi : 10.5802/aif.3415. http://www.numdam.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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