  We show that the tangent cone at the identity is not a complete
quasiconformal invariant for sub-Riemannian nilpotent groups. Namely, we show
that there exists a nilpotent Lie group equipped with left invariant
sub-Riemannian metric that is not locally quasiconformally equivalent to its
tangent cone at the identity. In particular, such spaces are not locally
bi-Lipschitz homeomorphic. The result is based on the study of Carnot groups
that are rigid in the sense that their only quasiconformal maps are the
translations and the dilations.
