Commensurating actions of birational groups and groups of pseudo-automorphisms

Cantat, Serge; de Cornulier, Yves

Journal de l’École polytechnique — Mathématiques, Tome 6 (2019), pp. 767-809.

Les pseudo-automorphismes sont les transformations birationnelles qui sont régulières en codimension $1$. On emploie des idées de théorie géométrique des groupes pour obtenir qu’un groupe de transformations birationnelles satisfaisant une propriété de point fixe sur les complexes cubiques CAT(0), par exemple un groupe ayant la propriété (T) de Kazhdan, est birationnellement conjugué à un groupe agissant par pseudo-automorphismes sur un ouvert de Zariski non vide. On utilise cet argument pour classifier, modulo conjugaison birationnelle, les groupes de transformations birationnelles de surfaces avec cette propriété de point fixe.

Pseudo-automorphisms are birational transformations acting as regular automorphisms in codimension $1$. We import ideas from geometric group theory to prove that a group of birational transformations that satisfies a fixed point property on cat$\left(0\right)$ cubical complexes, for example a discrete group with Kazhdan Property (T), is birationally conjugate to a group acting by pseudo-automorphisms on some non-empty Zariski-open subset. We apply this argument to classify groups of birational transformations of surfaces with this fixed point property up to birational conjugacy.

Classification : 14E07, 14J50, 20F65
Keywords: Cremona group, birational group, commensurating action, algebraic surfaces, regularization
Mot clés : Groupe de Cremona, groupe birationnel, action commensurante, surfaces algébriques, régularisation

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