  We prove that any action of a higher rank lattice on a Gromov-hyperbolic
space is elementary. More precisely, it is either elliptic or parabolic. This
is a large generalization of the fact that any action of a higher rank lattice
on a tree has a fixed point. A consequence is that any quasi-action of a higher
rank lattice on a tree is elliptic, i.e. it has Manning's property (QFA).
Moreover, we obtain a new proof of the theorem of Farb-Kaimanovich-Masur that
any morphism from a higher rank lattice to a mapping class group has finite
image, without relying on the Margulis normal subgroup theorem nor on bounded
cohomology. More generally, we prove that any morphism from a higher rank
lattice to a hierarchically hyperbolic group has finite image. In the Appendix,
Vincent Guirardel and Camille Horbez deduce rigidity results for morphisms from
a higher rank lattice to various outer automorphism groups.
