Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



The decomposition group of a line in the plane

Authors: Isac Hedén and Susanna Zimmermann
Journal: Proc. Amer. Math. Soc. 145 (2017), 3665-3680
MSC (2010): Primary 14E07
Published electronically: May 24, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [AC2002] Maria Alberich-Carramiñana, Geometry of the plane Cremona maps, Lecture Notes in Mathematics, vol. 1769, Springer-Verlag, Berlin, 2002. MR 1874328
  • [Bla2012] Jérémy Blanc, Simple relations in the Cremona group, Proc. Amer. Math. Soc. 140 (2012), no. 5, 1495-1500. MR 2869134,
  • [BPV2009] Jérémy Blanc, Ivan Pan, and Thierry Vust, On birational transformations of pairs in the complex plane, Geom. Dedicata 139 (2009), 57-73. MR 2481837,
  • [Cas1901] Guido Castelnuovo, Le trasformazioni generatrici del gruppo cremoniano nel piano, Atti della R. Accad. delle Scienze di Torino 36 (1901), 861-874.
  • [Cis2001] Ian Chiswell, Introduction to $ \Lambda $-trees, World Scientific Publishing Co., Inc., River Edge, NJ, 2001. MR 1851337
  • [Cor2013] 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,
  • [EJZ010] Mikhail Ershov and Andrei Jaikin-Zapirain, Property (T) for noncommutative universal lattices, Invent. Math. 179 (2010), no. 2, 303-347. MR 2570119,
  • [Ser1977] Jean-Pierre Serre, Arbres, amalgames, $ {\rm SL}_{2}$, Société Mathématique de France, Paris, 1977 (French). Avec un sommaire anglais; Rédigé avec la collaboration de Hyman Bass; Astérisque, No. 46. MR 0476875
  • [Wat1982] Yasuo Watatani, Property T of Kazhdan implies property FA of Serre, Math. Japon. 27 (1982), no. 1, 97-103. MR 649023

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 14E07

Retrieve articles in all journals with MSC (2010): 14E07

Additional Information

Isac Hedén
Affiliation: Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502 Japan

Susanna Zimmermann
Affiliation: Departement Mathematik und Informatik, Universität Basel, Spiegelgasse 1, 4051 Basel, Switzerland

Received by editor(s): January 29, 2016
Received by editor(s) in revised form: April 21, 2016
Published electronically: May 24, 2017
Additional Notes: The first-named author is an International Research Fellow of the Japanese Society for the Promotion of Sciences, and this work was supported by Grant-in-Aid for JSPS Fellows Number 15F15751. Both authors acknowledge support by the Swiss National Science Foundation Grant “Birational Geometry” PP00P2_153026.
Communicated by: Lev Borisov
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society