The decomposition group of a line in the plane

Hedén, Isac; Zimmermann, Susanna

doi:10.1090/proc/13263

The decomposition group of a line in the plane
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by Isac Hedén and Susanna Zimmermann PDF

Proc. Amer. Math. Soc. 145 (2017), 3665-3680 Request permission

Abstract:

We show that the decomposition group of a line $L$ in the plane, i.e., the subgroup of plane birational transformations that send $L$ to itself birationally, is generated by its elements of degree 1 and one element of degree 2, and that it does not decompose as a non-trivial amalgamated product.

References

Maria Alberich-Carramiñana, Geometry of the plane Cremona maps, Lecture Notes in Mathematics, vol. 1769, Springer-Verlag, Berlin, 2002. MR 1874328, DOI 10.1007/b82933
Jérémy Blanc, Simple relations in the Cremona group, Proc. Amer. Math. Soc. 140 (2012), no. 5, 1495–1500. MR 2869134, DOI 10.1090/S0002-9939-2011-11004-9
Jérémy Blanc, Ivan Pan, and Thierry Vust, On birational transformations of pairs in the complex plane, Geom. Dedicata 139 (2009), 57–73. MR 2481837, DOI 10.1007/s10711-008-9316-3
Guido Castelnuovo, Le trasformazioni generatrici del gruppo cremoniano nel piano, Atti della R. Accad. delle Scienze di Torino 36 (1901), 861–874.
Ian Chiswell, Introduction to $\Lambda$-trees, World Scientific Publishing Co., Inc., River Edge, NJ, 2001. MR 1851337, DOI 10.1142/4495
Serge Cantat and Stéphane Lamy, Normal subgroups in the Cremona group, Acta Math. 210 (2013), no. 1, 31–94. With an appendix by Yves de Cornulier. MR 3037611, DOI 10.1007/s11511-013-0090-1
Mikhail Ershov and Andrei Jaikin-Zapirain, Property (T) for noncommutative universal lattices, Invent. Math. 179 (2010), no. 2, 303–347. MR 2570119, DOI 10.1007/s00222-009-0218-2
Jean-Pierre Serre, Arbres, amalgames, $\textrm {SL}_{2}$, Astérisque, No. 46, Société Mathématique de France, Paris, 1977 (French). Avec un sommaire anglais; Rédigé avec la collaboration de Hyman Bass. MR 0476875
Yasuo Watatani, Property T of Kazhdan implies property FA of Serre, Math. Japon. 27 (1982), no. 1, 97–103. MR 649023