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

## TL;DR

This paper investigates the rate at which Cayley graphs of non-singular 2-step nilpotent groups converge to their asymptotic cone, establishing an optimal convergence speed of O(1/n) under certain geometric conditions.

## Contribution

It proves that non-singular 2-step nilpotent groups have a convergence speed of O(1/n) to their asymptotic cone, extending previous results and linking geometric properties to convergence rates.

## Key findings

- Convergence speed is O(1/n) for non-singular 2-step nilpotent groups.
- Absence of abnormal geodesics characterizes non-singular cases in subFinsler geometry.
- Results generalize known convergence rates for abelian groups to certain non-abelian 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.