Contact and 1-quasiconformal maps on Carnot groups

This paper answers to some questions that remained open for some time in the community of mathematicians working on quasiconformal mapping theory in subriemannian geometry. The first result presented here is the characterisation of the rigidity of Carnot groups in the class of C2 contact maps, obtained by extending Tanaka theory from its classical domain of C∞ contact vector fields to the pseudogroup of local C2 contact mappings. The second result is a Liouville type theorem proved for all Carnot groups other than R or R2. The proof rests upon recent results of Capogna and Cowling and classical results on prolonging the conformal Lie algebra. As an additional goal, this article aims to provide a partial exposition on Tanaka theory and to give an elementary proof of a key result due to Guellimen, Quillen and Sternberg concerning complex characteristics.