Ultrarigid tangents of sub-Riemannian nilpotent groups
  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. 
Enrico Le Donne, Alessandro Ottazzi, Ben Warhurst
math.MG math.AP math.DG math.GR