Hyperbolicity for large automorphism groups of projective surfaces

Serge Cantat; Romain Dujardin

doi:10.5802/jep.294

Hyperbolicity for large automorphism groups of projective surfaces
[Hyperbolicité pour les groupes d’automorphismes des surfaces projectives]

Serge Cantat ¹ ; Romain Dujardin ²

¹ IRMAR, Campus de Beaulieu, bât. 22-23, 263 avenue du Général Leclerc, CS 74205, 35042, Rennes Cedex
² Sorbonne Université and Université Paris Cité, Laboratoire de Probabilités, Statistique et Modélisation (LPSM), F-75005 Paris, France

Journal de l’École polytechnique — Mathématiques, Tome 12 (2025), pp. 421-480.

Résumé
Abstract

Nous étudions les propriétés d’hyperbolicité de l’action des groupes non élémentaires d’automorphismes des surfaces complexes compactes, notamment des surfaces K3 et d’Enriques. Si un tel groupe contient un élément parabolique, nous montrons que toute mesure invariante Zariski-diffuse admet un exposant de Lyapunov non nul. Avec nos travaux antérieurs, on en déduit des critères simples pour l’expansion uniforme de la dynamique aléatoire, que le groupe contienne des éléments paraboliques ou non. Cette propriété d’expansion a des conséquences dynamiques importantes : classification des adhérences des orbites, équidistribution, propriétés d’ergodicité, etc. Nous en profitons pour faire le point sur la notion d’expansion uniforme pour les actions de groupes discrets de difféomorphismes de variétés compactes et la construction de fonctions de Margulis, ceci sous des hypothèses de moment optimales.

We study the hyperbolicity properties of the action of a non-elementary automorphism group on a compact complex surface, with an emphasis on K3 and Enriques surfaces. A first result is that when such a group contains parabolic elements, Zariski diffuse invariant measures automatically have non-zero Lyapunov exponents. In combination with our previous work, this leads to simple criteria for a uniform expansion property on the whole surface, for groups with and without parabolic elements. This, in turn, has strong consequences on the dynamics: description of orbit closures, equidistribution, ergodicity properties, etc. Along the way, we provide a reference discussion on uniform expansion of non-linear discrete group actions on compact (real) manifolds and the construction of Margulis functions under optimal moment conditions.

Reçu le : 2022-11-15
Accepté le : 2025-02-28
Publié le : 2025-03-07

DOI : 10.5802/jep.294

Classification : 37F80, 37H15, 37C40, 14J50
Keywords: Uniform expansion, Margulis function, invariance principle, random holomorphic dynamics, automorphisms of K3 surfaces
Mots-clés : Expansion uniforme, fonction de Margulis, principe d’invariance, dynamique holomorphe aléatoire, automorphismes des surfaces K3

Affiliations des auteurs :

Serge Cantat ¹ ; Romain Dujardin ²

¹ IRMAR, Campus de Beaulieu, bât. 22-23, 263 avenue du Général Leclerc, CS 74205, 35042, Rennes Cedex
² Sorbonne Université and Université Paris Cité, Laboratoire de Probabilités, Statistique et Modélisation (LPSM), F-75005 Paris, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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Serge Cantat; Romain Dujardin. Hyperbolicity for large automorphism groups of projective surfaces. Journal de l’École polytechnique — Mathématiques, Tome 12 (2025), pp. 421-480. doi : 10.5802/jep.294. https://jep.centre-mersenne.org/articles/10.5802/jep.294/

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