Twisted cocycles and rigidity problems

We consider a class of cohomologies associated to a group action, outline a duality method for their calculation, and apply it to study different questions related to the group action. In particular, we prove a number of results on infinitesimal and cohomological rigidity of higher rank cocompact lattice actions on imaginary boundaries of some symmetric spaces (as well as results on cohomologies of some partially hyperbolic actions and lattice actions on a broader class of homogeneous spaces). We also obtain a very transparent proof of local $C^3$ rigidity of projective actions of cocompact lattices in $PSL(2,\Bbb{R})$.