Hyperbolic rigidity of higher rank lattices

We will show that every action by isometries of a higher rank lattice on a Gromov-hyperbolic space is elementary. Among consequences, we obtain another proof of the Farb-Kaimanovich-Masur Theorem that any morphism from a higher rank lattice to a mapping class group has finite image. Guirardel and Horbez also deduce another proof of the Bridson-Wade Theorem that any morphism from a higher rank lattice to Out(Fn) has finite image.​