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

摘要

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) of for nilpotent groups of class and 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 for the volume of Cayley balls of a general nilpotent group of class r. We also state a number of related conjectural statements.