Finitely generated subgroups of algebraic elements of plane Cremona groups are bounded
[Les sous-groupes de type fini d’éléments algébriques du groupe de Cremona planaire sont bornés]
Lonjou, Anne1 ; Przytycki, Piotr2 ; Urech, Christian3
1Department of Mathematics, University of the Basque Country UPV/EHU, Sarriena s/n, 48940 Leioa, Bizkaia, Spain & IKERBASQUE, Basque Foundation for Science, Bilbao, Spain
2Department of Mathematics and Statistics, McGill University, Burnside Hall, 805 Sherbrooke Street West, Montreal, QC, H3A 0B9, Canada
3Department of Mathematics, ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland
Journal de l’École polytechnique — Mathématiques, Tome 11 (2024), pp. 1011-1028

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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/jep.271
Classification : 14E07, 20F65, 37F10
Keywords: Cremona group, algebraic elements, locally elliptic actions, degree growth
Mots-clés : Groupe de Cremona, éléments algébriques, actions purement elliptiques, croissance des degrés

Lonjou, Anne  1   ; Przytycki, Piotr  2   ; Urech, Christian  3

1 Department of Mathematics, University of the Basque Country UPV/EHU, Sarriena s/n, 48940 Leioa, Bizkaia, Spain & IKERBASQUE, Basque Foundation for Science, Bilbao, Spain
2 Department of Mathematics and Statistics, McGill University, Burnside Hall, 805 Sherbrooke Street West, Montreal, QC, H3A 0B9, Canada
3 Department of Mathematics, ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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     title = {Finitely generated subgroups of algebraic~elements of plane {Cremona} groups are~bounded},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
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Lonjou, Anne; Przytycki, Piotr; Urech, Christian. Finitely generated subgroups of algebraic elements of plane Cremona groups are bounded. Journal de l’École polytechnique — Mathématiques, Tome 11 (2024), pp. 1011-1028. doi: 10.5802/jep.271

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