On the rate of convergence to the asymptotic cone for nilpotent groups and subFinsler geometry

TL;DR

This paper investigates the rate at which finitely generated nilpotent groups with word metrics converge to their asymptotic cones, establishing a lower bound on the convergence rate and deriving volume error estimates.

Contribution

It provides a quantitative rate of convergence in Gromov-Hausdorff metric for nilpotent groups, addressing Gromov's question and highlighting the impact of abnormal geodesics.

Findings

Convergence rate cannot be better than n^{1/2} for non-abelian nilpotent groups.

Derived an error term for volume of Cayley balls: O(n^{d-2/(3r)}).

Established a lower bound on convergence rate related to abnormal geodesics.

Abstract

Addressing a question of Gromov, we give a rate in Pansu's theorem about the convergence in Gromov-Hausdorff metric of a finitely generated nilpotent group equipped with a left-invariant word metric scaled by a factor 1/n towards its asymptotic cone. We show that due to the possible presence of abnormal geodesics in the asymptotic cone, this rate cannot be better than n^{1/2} for general non-abelian nilpotent groups. As a corollary we also get an error term of the form vol(B(n))=cn^d + O(n^{d-2/(3r)}) for the volume of Cayley balls of a nilpotent group with nilpotency class r. We also state a number of related conjectural statements.