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This post shows that every Carnot group is a doubling metric space. However, what is its doubling constant?

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 ...