Subgroups of elliptic elements of the Cremona group
-
Christian Urech
Abstract
The Cremona group is the group of birational transformations of the complex projective plane. In this paper we classify its subgroups that consist only of elliptic elements using elementary model theory. This yields in particular a description of the structure of torsion subgroups. As an application, we prove the Tits alternative for arbitrary subgroups of the Cremona group, generalizing a result of Cantat. We also describe solvable subgroups of the Cremona group and their derived length, refining results from Déserti.
Funding statement: The author gratefully acknowledges support by the Swiss National Science Foundation Grant “Birational Geometry” PP00P2 128422 /1 as well as by the Geldner-Stiftung, the FAG Basel, the Janggen Pöhn-Stiftung and the State Secretariat for Education, Research and Innovation of Switzerland.
Acknowledgements
I express my warmest thanks to my PhD-advisors Jérémy Blanc and Serge Cantat for sharing their beautiful view on the Cremona group with me and for their constant support and helpful guidance. I also thank Michel Brion, Ivan Cheltsov, and Julie Déserti for many helpful comments. Special thanks go to the referee whose valuable comments have significantly improved the paper.
References
[1] A. Beauville, Complex algebraic surfaces, 2nd ed., London Math. Soc. Stud. Texts 34, Cambridge University, Cambridge 1996. 10.1017/CBO9780511623936Search in Google Scholar
[2]
M. Bestvina, M. Feighn and M. Handel,
The Tits alternative for
[3]
A. Biał ynicki Birula,
Remarks on the action of an algebraic torus on
[4] J. Blanc, Sous-groupes algébriques du groupe de Cremona, Transform. Groups 14 (2009), no. 2, 249–285. 10.1007/s00031-008-9046-5Search in Google Scholar
[5] J. Blanc and S. Cantat, Dynamical degrees of birational transformations of projective surfaces, J. Amer. Math. Soc. 29 (2016), no. 2, 415–471. 10.1090/jams831Search in Google Scholar
[6] J. Blanc and J.-P. Furter, Topologies and structures of the Cremona groups, Ann. of Math. (2) 178 (2013), no. 3, 1173–1198. 10.4007/annals.2013.178.3.8Search in Google Scholar
[7] M. Brion, Notes on automorphism groups of projective varieties, Notes de l’école d’été “Varieties and Group Actions” (2018), http://www-fourier.univ-grenoble-alpes.fr/~mbrion/autos_final.pdf. Search in Google Scholar
[8] S. Cantat, Sur les groupes de transformations birationnelles des surfaces, Ann. of Math. (2) 174 (2011), no. 1, 299–340. 10.4007/annals.2011.174.1.8Search in Google Scholar
[9] S. Cantat, Sur les groupes de transformations birationnelles des surfaces, Lecture Notes (2011), https://perso.univ-rennes1.fr/serge.cantat/Articles/cremona_long.pdf. 10.4007/annals.2011.174.1.8Search in Google Scholar
[10] S. Cantat, The Cremona groups, Proc. Sympos. Pure Math. 97 (2018), 10.1090/pspum/097.1/01690. 10.1090/pspum/097.1/01690Search in Google Scholar
[11] S. Cantat and I. Dolgachev, Rational surfaces with a large group of automorphisms, J. Amer. Math. Soc. 25 (2012), no. 3, 863–905. 10.1090/S0894-0347-2012-00732-2Search in Google Scholar
[12] S. Cantat and S. Lamy, Normal subgroups in the Cremona group, Acta Math. 210 (2013), no. 1, 31–94. 10.1007/s11511-013-0090-1Search in Google Scholar
[13] M. Clay and D. Margalit, Groups, Office hours with a geometric group theorist, Princeton University, Princeton (2017), 3–20. 10.2307/j.ctt1vwmg8g.5Search in Google Scholar
[14] C. W. Curtis and I. Reiner, Representation theory of finite groups and associative algebras, John Wiley & Sons, New York 1962. Search in Google Scholar
[15] P. de la Harpe, Topics in geometric group theory, Chicago Lectures in Math., University of Chicago, Chicago 2000. Search in Google Scholar
[16] T. Delzant and P. Py, Kähler groups, real hyperbolic spaces and the Cremona group, Compos. Math. 148 (2012), no. 1, 153–184. 10.1112/S0010437X11007068Search in Google Scholar
[17] M. Demazure, Sous-groupes algébriques de rang maximum du groupe de Cremona, Ann. Sci. Éc. Norm. Supér. (4) 3 (1970), 507–588. 10.24033/asens.1201Search in Google Scholar
[18] J. Déserti, On solvable subgroups of the Cremona group, Illinois J. Math. 59 (2015), no. 2, 345–358. 10.1215/ijm/1462450705Search in Google Scholar
[19] J. Diller and C. Favre, Dynamics of bimeromorphic maps of surfaces, Amer. J. Math. 123 (2001), no. 6, 1135–1169. 10.1353/ajm.2001.0038Search in Google Scholar
[20] I. V. Dolgachev, Classical algebraic geometry. A modern view, Cambridge University, Cambridge 2012. 10.1017/CBO9781139084437Search in Google Scholar
[21] I. V. Dolgachev and V. A. Iskovskikh, Finite subgroups of the plane Cremona group, Algebra, arithmetic, and geometry: In honor of Y. I. Manin. Vol. I, Progr. Math. 269, Birkhäuser, Boston (2009), 443–548. 10.1007/978-0-8176-4745-2_11Search in Google Scholar
[22] F. Enriques, Sui gruppi continui di trasformazioni cremoniane nel piano, Rom. Accad. L. Rend. 5 (1893), 468–473. Search in Google Scholar
[23] C. Favre, Le groupe de Cremona et ses sous-groupes de type fini, Séminaire Bourbaki. Volume 2008/2009. Exposés 997–1011 Astérisque 332, Société Mathématique de France, Paris (2010), 11–13, Exp. No. 998. Search in Google Scholar
[24] J.-P. Furter and P.-M. Poloni, On the maximality of the triangular subgroup, Ann. Inst. Fourier (Grenoble) 68 (2018), no. 1, 393–421. 10.5802/aif.3165Search in Google Scholar
[25] M. H. Gizatullin, Rational G-surfaces, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), no. 1, 110–144, 239. Search in Google Scholar
[26] J. Grivaux, Parabolic automorphisms of projective surfaces (after M. H. Gizatullin), Mosc. Math. J. 16 (2016), no. 2, 275–298. 10.17323/1609-4514-2016-16-2-275-298Search in Google Scholar
[27] M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ. 8, Springer, New York, (1987), 75–263. 10.1007/978-1-4613-9586-7_3Search in Google Scholar
[28] J. E. Humphreys, Linear algebraic groups, Grad. Texts in Math. 21, Springer, New York, 1975. 10.1007/978-1-4684-9443-3Search in Google Scholar
[29] V. A. Iskovskikh and I. R. Shafarevich, Algebraic surfaces, Algebraic geometry. II, Encyclopaedia Math. Sci. 35, Springer, Berlin (1996), 127–262. 10.1007/978-3-642-60925-1_2Search in Google Scholar
[30] N. V. Ivanov, Algebraic properties of the Teichmüller modular group, Dokl. Akad. Nauk SSSR 275 (1984), no. 4, 786–789. Search in Google Scholar
[31] O. Karpenkov, Geometry of continued fractions, Algorithms Comput. Math. 26, Springer, Heidelberg 2013. 10.1007/978-3-642-39368-6Search in Google Scholar
[32]
S. Lamy,
Dynamique des groupes paraboliques d’automorphismes polynomiaux de
[33]
S. Lamy,
L’alternative de Tits pour
[34] A. I. Mal’cev, On the faithful representation of infinite groups by matrices, Mat. Sb 8 (1940), no. 50, 405–422. 10.1090/trans2/045/01Search in Google Scholar
[35] Y. I. Manin, Cubic forms, 2nd ed., North-Holland Math. Libr. 4, North-Holland, Amsterdam 1986. Search in Google Scholar
[36] M. F. Newman, The soluble length of soluble linear groups, Math. Z. 126 (1972), 59–70. 10.1007/BF01580356Search in Google Scholar
[37] N. I. Shepherd-Barron, Some effectivity questions for plane cremona transformations, preprint (2013), https://arxiv.org/abs/1311.6608. Search in Google Scholar
[38] J. Tits, Free subgroups in linear groups, J. Algebra 20 (1972), 250–270. 10.1016/0021-8693(72)90058-0Search in Google Scholar
[39] H. Umemura, On the maximal connected algebraic subgroups of the Cremona group. I, Nagoya Math. J. 88 (1982), 213–246. 10.1017/S0027763000020183Search in Google Scholar
[40] C. Urech, Subgroups of cremona groups, Ph.D. thesis, University of Basel/ University of Rennes 1, 2017. Search in Google Scholar
[41] A. Weil, On algebraic groups of transformations, Amer. J. Math. 77 (1955), 355–391. 10.1007/978-1-4757-1705-1_73Search in Google Scholar
[42]
D. Wright,
Abelian subgroups of
© 2020 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Eichler cohomology and zeros of polynomials associated to derivatives of L-functions
- Subgroups of elliptic elements of the Cremona group
- On derived equivalences and homological dimensions
- Area minimizing surfaces of bounded genus in metric spaces
- Half-space theorems for the Allen–Cahn equation and related problems
- Stability conditions on product varieties
- Uniform bounds on harmonic Beltrami differentials and Weil–Petersson curvatures
- Bergman–Einstein metrics, a generalization of Kerner’s theorem and Stein spaces with spherical boundaries
- On birational boundedness of foliated surfaces
Articles in the same Issue
- Frontmatter
- Eichler cohomology and zeros of polynomials associated to derivatives of L-functions
- Subgroups of elliptic elements of the Cremona group
- On derived equivalences and homological dimensions
- Area minimizing surfaces of bounded genus in metric spaces
- Half-space theorems for the Allen–Cahn equation and related problems
- Stability conditions on product varieties
- Uniform bounds on harmonic Beltrami differentials and Weil–Petersson curvatures
- Bergman–Einstein metrics, a generalization of Kerner’s theorem and Stein spaces with spherical boundaries
- On birational boundedness of foliated surfaces
