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My question is in some sense a less ambitious version of the following MO question where the answer was inconclusive. A Carnot group of step $N$ can be identified within the tensor algebra, modulo ...

Background: Let the smooth vector fields $X=(X_1,\cdots,X_m)$ define on $\mathbb{R}^n$ and they satisfy the following assumption: (H1): There is a dilation structure $$\delta_{t}:\mathbb{R}^n\to \...

Note that $(\mathbb{Z}^2,d_W)$ where $d_W$ is word metric has asymptotic cone $$(\mathbb{R}^2,\| \ \|_1)=\lim_{t>0\rightarrow 0}\ t(\mathbb{Z}^2,d_W)$$ And Heisenberg group $\mathbb{H}^3$ has an ...