Ultrarigid tangents of sub-Riemannian nilpotent groups

TL;DR

This paper demonstrates that the tangent cone at the identity does not serve as a complete quasiconformal invariant for sub-Riemannian nilpotent groups, revealing limitations in their geometric classification.

Contribution

It constructs examples of nilpotent Lie groups with sub-Riemannian metrics that are not locally quasiconformally equivalent to their tangent cones, highlighting new rigidity phenomena.

Findings

Existence of nilpotent groups not locally quasiconformally equivalent to their tangent cones

Identification of rigid Carnot groups with only trivial quasiconformal maps

Counterexamples to tangent cone as a complete invariant

Abstract

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.