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?

1 Answer

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 ).

Every Carnot group has a "snowflake" embedding into some Euclidean space, by Assouad's embedding theorem. (see https://en.wikipedia.org/wiki/Doubling_space ). Such an embedding is quasiconformal according to the so-called "metric definition", but not the "geometric definition". (See the article of Heinonen above for these terms.)

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...