Ultrarigid tangents of sub-Riemannian nilpotent groups (Q486841)

The main goal of this paper is to show that the converse of the theorem of Margulis and Mostow fails in a strong sense. More precisely, for the class of sub-Riemannian nilpotent groups, the authors show that the tangent cone at the identity is not a complete quasiconformal invariant. Namely, it is shown 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 holomorphic. 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.