On quasi-isometric invariants: Rigidity and related phenomena

We investigate quasi-isometric invariants that are outgrowths of extensions of Mostow\u27s strong rigidity. In particular, we extend a theorem of Hamenstadt, proving the rigidity of pinched negatively curved manifolds whose deck groups acting on their universal covers satisfy the duality condition and that have higher hyperbolic rank. Also, we consider quasi-isometric embeddings of nonabelian nilpotent Lie groups and construct a new invariant for them. We use this invariant to prove that there do not exist quasi-isometric embeddings of a nonabelian nilpotent Lie group into a space of nonpositive curvature