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 a finitely generated nilpotent group endowed with a word metric. The first result on this theme is given by Burago who showed that an abelian group endowed with a word metric converges to the normed space with the speed \(O\left( \frac{1}{n}\right) \) in the sense of Gromov–Hausdorff distance. Later Krat showed the same statement for the Heisenberg group, and Breuillard and Le Donne constructed an example, the direct product of the \(\mathbb {Z}\) and the Heisenberg group with a specific word metric, whose speed of convergence is precisely \(O\left( \frac{1}{\sqrt{n}}\right) \). For 2-step nilpotent groups, we show that if the asymptotic cone is non-singular, then the speed of convergence is \(O\left( \frac{1}{n}\right) \) for any choice of generating set. Our argument can be applied to every nilpotent Lie group with a left-invariant sub-Finsler metric. In terms of sub-Finsler geometry, the condition being non-singular is equivalent to the strongly bracket generating condition, and also to absence of abnormal curves.