Quasiisometries of negatively curved homogeneous manifolds associated with Heisenberg groups

We study quasiisometries between negatively curved homogeneous manifolds associated with diagonalizable derivations on Heisenberg algebras. We classify these manifolds up to quasiisometry, and show that all quasiisometries between such manifolds (except when they are complex hyperbolic spaces) are almost similarities. We prove these results by studying the quasisymmetric maps on the ideal boundary of these manifolds.