  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.
