On the speed of convergence to the asymptotic cone for non-singular nilpotent groups

Abstract

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.