Kähler groups, real hyperbolic spaces and the Cremona group. With an appendix by Serge Cantat

Generalizing a classical theorem of Carlson and Toledo, we prove that any Zariski dense isometric action of a Kähler group on the real hyperbolic space of dimension at least three factors through a homomorphism onto a cocompact discrete subgroup of PSL 2 (ℝ). We also study actions of Kähler groups on infinite-dimensional real hyperbolic spaces, describe some exotic actions of PSL 2 (ℝ) on these spaces, and give an application to the study of the Cremona group.