# quasi-conformal embedding of Carnot group into euclidean space

By Pansu's theorem, there are no bi-Lipschitz embeddings of Carnot groups (with exception of the Euclidean space itself) into Euclidean space. Do there exist quasi-conformal embeddings (into Eucl. sp.) of such groups?

It depends precisely what you mean by quasiconformal embedding.'' There are different definitions that do not agree in complete generality on all metric spaces. (See http://www.ams.org/notices/200611/whatis-heinonen.pdf ).
The above embedding will not be Pansu differentiable anywhere. Indeed, Pansu differentiability is the obstruction for bi-Lipschitz embeddings and also for nicer'' quasiconformal embeddings...