We prove that any finitely generated subgroup of the plane Cremona group consisting only of algebraic elements is of bounded degree. This follows from a more general result on ‘decent’ actions on infinite restricted products. We apply our results to describe the degree growth of finitely generated subgroups of the plane Cremona group.
Nous montrons que tout sous-groupe de type fini du groupe de Cremona planaire contenant seulement des éléments algébriques est de degré borné. Cela découle d’un résultat plus général sur les actions « décentes » sur les produits infinis restreints. Nous appliquons nos résultats pour décrire la croissance des degrés des sous-groupes de type fini du groupe de Cremona planaire.
Accepted:
Published online:
DOI: 10.5802/jep.271
Keywords: Cremona group, algebraic elements, locally elliptic actions, degree growth
Mots-clés : Groupe de Cremona, éléments algébriques, actions purement elliptiques, croissance des degrés
Anne Lonjou 1; Piotr Przytycki 2; Christian Urech 3
CC-BY 4.0
@article{JEP_2024__11__1011_0,
author = {Anne Lonjou and Piotr Przytycki and Christian Urech},
title = {Finitely generated subgroups of algebraic~elements of plane {Cremona} groups are~bounded},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {1011--1028},
publisher = {\'Ecole polytechnique},
volume = {11},
year = {2024},
doi = {10.5802/jep.271},
mrnumber = {4801142},
zbl = {07928809},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.271/}
}
TY - JOUR AU - Anne Lonjou AU - Piotr Przytycki AU - Christian Urech TI - Finitely generated subgroups of algebraic elements of plane Cremona groups are bounded JO - Journal de l’École polytechnique — Mathématiques PY - 2024 SP - 1011 EP - 1028 VL - 11 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.271/ DO - 10.5802/jep.271 LA - en ID - JEP_2024__11__1011_0 ER -
%0 Journal Article %A Anne Lonjou %A Piotr Przytycki %A Christian Urech %T Finitely generated subgroups of algebraic elements of plane Cremona groups are bounded %J Journal de l’École polytechnique — Mathématiques %D 2024 %P 1011-1028 %V 11 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.271/ %R 10.5802/jep.271 %G en %F JEP_2024__11__1011_0
Anne Lonjou; Piotr Przytycki; Christian Urech. Finitely generated subgroups of algebraic elements of plane Cremona groups are bounded. Journal de l’École polytechnique — Mathématiques, Volume 11 (2024), pp. 1011-1028. doi : 10.5802/jep.271. https://jep.centre-mersenne.org/articles/10.5802/jep.271/
[BC16] - “Dynamical degrees of birational transformations of projective surfaces”, J. Amer. Math. Soc. 29 (2016) no. 2, p. 415-471 | DOI | MR | Zbl
[BF13] - “Topologies and structures of the Cremona groups”, Ann. of Math. (2) 178 (2013) no. 3, p. 1173-1198 | DOI | MR | Zbl
[BF21] - “On the joint spectral radius for isometries of non-positively curved spaces and uniform growth”, Ann. Inst. Fourier (Grenoble) 71 (2021) no. 1, p. 317-391 | DOI | Numdam | MR | Zbl
[Can11] - “Sur les groupes de transformations birationnelles des surfaces”, Ann. of Math. (2) 174 (2011) no. 1, p. 299-340 | DOI | MR | Zbl
[Can18] - “The Cremona group”, in Algebraic geometry (Salt Lake City, 2015), Proc. Sympos. Pure Math., vol. 97.1, American Mathematical Society, Providence, RI, 2018, p. 101-142 | DOI | Zbl
[CD12] - “Rational surfaces with a large group of automorphisms”, J. Amer. Math. Soc. 25 (2012) no. 3, p. 863-905 | DOI | MR | Zbl
[Dan20] - “Degrees of iterates of rational maps on normal projective varieties”, Proc. London Math. Soc. (3) 121 (2020) no. 5, p. 1268-1310 | DOI | MR | Zbl
[DK18] - Geometric group theory, Amer. Math. Soc. Colloquium Publ., vol. 63, American Mathematical Society, Providence, RI, 2018 | DOI | MR
[Fav10] - “Le groupe de Cremona et ses sous-groupes de type fini”, in Séminaire Bourbaki. Volume 2008/2009, Astérisque, vol. 332, Société Mathématique de France, Paris, 2010, p. 11-43, Exp. No. 998 | Numdam | MR | Zbl
[Giz80] - “Rational -surfaces”, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980) no. 1, p. 110-144 | MR | Zbl
[GLU24] - “Cremona groups over finite fields, Neretin groups, and non-positively curved cube complexes”, Internat. Math. Res. Notices (2024) no. 1, p. 554-596 | DOI | MR | Zbl
[Gri16] - “Parabolic automorphisms of projective surfaces (after M. H. Gizatullin)”, Moscow Math. J. 16 (2016) no. 2, p. 275-298 | DOI | MR | Zbl
[HO21] - “Locally elliptic actions, torsion groups, and nonpositively curved spaces”, 2021 | arXiv
[Jun09] - The joint spectral radius: theory and applications, Lect. Notes in Control and Information Sci., vol. 385, Springer, Berlin, Heidelberg, 2009 | DOI | MR
[Kar24] - “Decency of group actions on restricted products” (2024), in preparation
[Lam24] - “The Cremona group” (2024), in preparation, available at https://www.math.univ-toulouse.fr/~slamy/blog/cremona.html
[LU21] - “Actions of Cremona groups on cube complexes”, Duke Math. J. 170 (2021) no. 17, p. 3703-3743 | MR | Zbl
[NOP22] - “Torsion groups do not act on 2-dimensional complexes”, Duke Math. J. 171 (2022) no. 6, p. 1379-1415 | MR | Zbl
[RS60] - “A note on the joint spectral radius”, Indag. Math. 22 (1960) no. 4, p. 379-381 | DOI | MR | Zbl
[Sch22] - “Relations in the Cremona group over a perfect field”, Ann. Inst. Fourier (Grenoble) 72 (2022) no. 1, p. 1-42 | DOI | Numdam | MR | Zbl
[Ser80] - Trees, Springer-Verlag, Berlin-New York, 1980 | DOI | MR
[Ser10] - “Le groupe de Cremona et ses sous-groupes finis”, in Séminaire Bourbaki. Volume 2008/2009, Astérisque, vol. 332, Société Mathématique de France, Paris, 2010, p. 75-100, Exp. No. 1000 | Numdam | MR | Zbl
[Ure17] - Subgroups of Cremona groups, Ph. D. Thesis, University of Basel, 2017
[Ure18] - “Remarks on the degree growth of birational transformations”, Math. Res. Lett. 25 (2018) no. 1, p. 291-308 | DOI | MR | Zbl
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