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 asymptotic cone. It is a subRiemannian metric. But what is aymptotic cone of its discrete group ?

Thank you in advance.