K\"ahler groups, real hyperbolic spaces and the Cremona group

Generalizing a classical theorem of Carlson and Toledo, we prove that any Zariski dense isometric action of a K"{a}hler group on the real hyperbolic space of dimension at least 3 factors through a homomorphism onto a cocompact discrete subgroup of PSL(2,R). We also study actions of K"{a}hler 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.