# What are Carnot groups?

I'm trying to learn the Pansu differentiability theorem and I need to know what Carnot groups are. Can someone please explain what Carnot groups are? An introductory reference would be greatly appreciated as well.

Thanks.

• What do you need to know that is not explained on the Wikipedia page (found easily by search engine, it would seem)? – Yemon Choi Jan 18 '15 at 15:22
• I guess I need an introductory reference with a bunch of examples. – Axiom Jan 18 '15 at 15:23
• Does this paper satisfy your needs? arxiv.org/abs/1304.7493 – Joonas Ilmavirta Jan 18 '15 at 16:08
• Yes thanks. That's what I was looking for. – Axiom Jan 18 '15 at 16:14
• From the algebraic point of view and a characterization among other nilpotent Lie groups, see arxiv.org/abs/1403.5295 (esp. 3.2 for the basic facts). – YCor Jan 18 '15 at 22:13

I learned the theory of Carnot group or more general subRiemannian manifolds from the thesis of Monti, which can be found here: http://www.math.unipd.it/~monti/PAPERS/TesiFinale.pdf In particular, Section 2 is devoted to the proof of Pansu-differentiability theorem.

Another good notes is by le donne, which can be found here . However the proof of Pansu differentiability theorem is stated without a proof.

By the way, the reference provided above "Gromov, Mikhael Carnot-Carathéodory spaces seen from within. Sub-Riemannian geometry, 79–323, Progr. Math., 144, Birkhäuser, Basel, 1996." is also a nice research suvery.

As an introductory reference, I'm a fan of the book Stratified Lie Groups and Potential Theory for their Sub-Laplacians by A. Bonfiglioli, E. Lanconelli and F. Uguzzoni.

MR1421823 Gromov, Mikhael Carnot-Carathéodory spaces seen from within. Sub-Riemannian geometry, 79–323, Progr. Math., 144, Birkhäuser, Basel, 1996.

• The post above would be interesting for your request: fabricebaudoin.wordpress.com/2013/01/05/… – Zbigniew Mar 30 '15 at 15:00
• @Zbignev: you entered your comment in the wrong place. You surely wanted to comment on the question, not on my answer. – Alexandre Eremenko Mar 30 '15 at 18:05
• Eremeko I do apologize, I'm not a frequent user of the mathoverflow. – Zbigniew Apr 1 '15 at 3:49