We provide the first examples of strongly dense representations of a hyperbolic 3-manifold group into S L (4, R) SL (4, R) and S U (3, 1) SU (3, 1) i. e. representations where every pair of non-commuting elements has Zariski dense image. Our examples are holonomy representations arising from projective deformations of its hyperbolic structure. As a corollary, we get that S L (4, R) SL (4, R) has non-Hitchin strongly dense surface subgroups.
Ricky Lee (Fri,) studied this question.