We study the speed of convergence to the asymptotic cone for Cayley graphs of nilpotent groups. Burago showed that $\{ (\mathbb{ Z}^d, \frac{ 1}{ n} \rho,id)\}_{ n\in\mathbb{ N}}$ converges to $(\mathbb{ R}^d,d_{ \infty},id)$ and its speed is $O(\frac{ 1}{ n})$ in the sense of Gromov-Hausdorff distance. Later Breuillard and Le Donne gave estimates for non-abelian cases, and constructed an example whose speed of convergence is slower than $O(\frac{ 1}{ \sqrt{ n}})$. For $2$-step nilpotent groups, we show that if the Mal'cev completion is non-singular, then the speed of convergence is $O(\frac{ 1}{ n})$ for any choice of generating set. In terms of subFinsler geometry, this condition is also equivalent to the absence of abnormal geodesics on the asymptotic cone.
Categories: math.GR