Rigidity of locally Hermitian symmetric rank one manifolds of infinite volume

We address G.D.Mostow, L.Bers and S.L.Krushkal questions on uniqueness of conformal/spherical CR structures on the sphere at infinity of non-compact (Hermitian) symmetric rank one spaces compatible with the action of a discrete isometry group. We construct such non-rigid discrete isometry groups whose quotients have infinite volumes and whose non-trivial deformations are induced by equivariant homeomorphisms of the (Hermitian) symmetric space with bounded horizontal distortion. This non-rigidity is related to non-ergodic dynamics of our discrete isometry group actions on the limit set which could be the whole sphere at infinity.