Commensurating actions of birational groups and groups of pseudo-automorphisms

Serge Cantat (1) , Yves de Cornulier (2)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 AGL - Algèbre, géométrie, logique

Pseudo-automorphisms are birational transformations acting as regular automorphisms in codimension 1. We import ideas from geometric group theory to study groups of birational transformations, and prove that a group of birational transformations that satisfies a fixed point property on CAT(0) cubical complexes is birationally conjugate to a group acting by pseudo-automorphisms on a 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.

Domaines: Géométrie algébrique [math.AG]; Théorie des groupes [math.GR]; Topologie géométrique [math.GT]

Serge Cantat, Yves de Cornulier. Commensurating actions of birational groups and groups of pseudo-automorphisms. Journal de l'École polytechnique — Mathématiques, 2019, 6, pp.767-809. ⟨10.5802/jep.106⟩. ⟨hal-01522744⟩