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

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 1/n . We obtain a convergence rate (measured in the Gromov-Hausdorff metric) of n -2/3r for nilpotent groups of class r > 2 and n-1/2 for 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 form vol(B(n))=cnd +O(nd-2/3r) for the volume of Cayley balls of a general nilpotent group of class r. We also state a number of related conjectural statements.