Asymptotics of Riemannian Lie groups with nilpotency step 2

We derive sharp estimates comparing asymptotic Riemannian or sub-Riemannian metrics in 2-step nilpotent Lie groups. For each metric, we construct a Carnot metric whose square remains at bounded distance from the square of the original metric. In particular, we deduce the analogue of a conjectire by Burago-Margulis: every 2-step nilpotent Riemannian Lie group is at bounded distance from its asymptotic cone. As a consequence, we obtain a refined estimate of the error term in the asymptotic expansion of the volume of the (sub-)Riemannian metric balls. To achive this, we develop a novel technique to efficiently perturb rectifiable curves modifying their endpoints in a prescribed vertical direction.