We study the variety of actions of a fixed (Chevalley) group on arbitrary geodesic, Gromov hyperbolic spaces. In high rank we obtain a complete classification. In rank one, we obtain some partial results and give a conjectural picture.
"Actions of certain arithmetic groups on Gromov hyperbolic spaces." Algebr. Geom. Topol. 8 (3) 1371 - 1402, 2008. https://doi.org/10.2140/agt.2008.8.1371