Local rigidity of boundary actions

We consider the question how hard/easy it is to perturb a nice action of a group on a manifold. In general, when the group is "large'', it is a general belief that the only way to perturb actions is by conjugating the actions by a map close to identity. This is indeed true in many situations, even perturbations are allowed to have low regularity. On the other hand, there are also cases, perturbations are flexible to obtain.
In this talk, we'll mostly look at actions of uniform lattices in semisimple Lie groups or more generally fundamental groups of nonpositively curved closed manifolds on geodesic boundaries and Furstenberg boundaries.