Numerical invariants and volume rigidity for hyperbolic lattices

We prove a generalization of Mostow-Prasad rigidity by showing that the volume function on the PO(m,1)-character variety of a non-uniform real hyperbolic lattice of PO(p,1) stays away from its maximum outside a suitable analytic neighborhood of the class of the discrete and faithful representation, when m>=p>=3. The same for non-uniform complex and quaternionic hyperbolic lattices for m>=p>=2. When G is a non-uniform lattice of PSL(2,C) without torsion we define the omega-Borel invariant for representations into SL(n,C_om) and we discuss its properties.