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.
2026-04-19 03:42:26.1776570146
What are Carnot groups?
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A Carnot group is a special type of Lie group $\mathcal{G} $ that is nilpotent (almost abelian) and simply connected. Its Lie algebra $\mathcal{g}$ gives a nilpotent stratification consisting of $k$ steps. This can be represented in the form $\mathcal{V}_1\oplus\cdots \oplus \mathcal{V}_k$ with $[\mathcal{V}_1,\mathcal{V}_m]=\mathcal{V}_{1+m},~m=1,\cdots ,k-1$. For example, the space $\mathbb{R}^n$ under the binary operation of addition is a commutative Carnot group. Carnot groups possess a Carnot-Caratheodory metric (a metric space with integer Hausdorff dimension greater than its dimension) and Pansu derivative (a special type of derivative adapted to a Carnot group).