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

Abstract

Addressing a question of Gromov, we give a rate in Pansu’s theorem about the convergence to the asymptotic cone of a finitely generated nilpotent group equipped with a left-invariant word metric rescaled by a factor. We obtain a convergence rate (measured in the Gromov–Hausdorff metric) offor nilpotent groups of classandfor nilpotent groups of class 2. We also show that the latter result is sharp, and we make a connection between this sharpness and the presence of so-called abnormal geodesics in the asymptotic cone. As a corollary, we get an error term of the formfor the volume of Cayley balls of a general nilpotent group of classr. We also state a number of related conjectural statements.