Instead of hyperbolic manifolds, I suggest you to follow smooth Riemannian manifolds, i.e., analytic manifolds, such as Stiefel and Grassmann manifolds are much better than hyperbolic manifolds.

A good question! A partial answer is that, in general, the universal covering space of such a manifold admits actions of hyperbolic groups (in the sense of M.Gromov), having "a rich structure". + https://arxiv.org/pdf/1409.2155.pdf