Conformal and CR mappings on Carnot groups

TL;DR

This paper studies conformal and CR mappings on stratified Lie groups with CR structures, showing their equivalence and characterizing their form on product groups, revealing structural rigidity and classification.

Contribution

It establishes the equivalence of conformal and CR/anti-CR maps on certain Lie groups and characterizes their structure on product groups, extending understanding of geometric mappings.

Findings

Conformal maps coincide with CR and anti-CR diffeomorphisms.

On product groups, CR and anti-CR maps are essentially product maps.

CR and anti-CR maps are affine in each component, up to permutation.

Abstract

We consider a class of stratified groups with a CR structure and a compatible control distance. For these Lie groups we show that the space of conformal maps coincide with the space of CR and anti-CR diffeomorphisms. Furthermore, we prove that on products of such groups, all CR and anti-CR maps are product maps, up to a permutation isomorphism, and affine in each component.