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October 13, 2025Open Access

Asymptotics of Riemannian Lie groups with nilpotency step 2

EDEnrico Le DonneUniversity of Fribourg LNLuca NalonUniversity of Fribourg SGSebastiano Nicolussi GoloUniversity of Padua

Key Points

Refined estimates for error terms in volume expansions of (sub-)Riemannian metrics were established, enhancing understanding in geometric analysis.
The study shows how a Carnot metric maintains bounded distance from original sub-Riemannian metrics, crucial for accurate comparisons.
A novel technique for perturbing sub-Riemannian geodesics was developed, allowing specific modifications of endpoints for improved accuracy.
The findings highlight implications for structural properties of nilpotent Lie groups, inviting further exploration of asymptotic metrics.

Abstract

We derive estimates comparing asymptotic Riemannian or sub-Riemannian metrics in step-2 nilpotent Lie groups. Given a sub-Riemannian metric, we construct a Carnot metric whose square remains at a bounded distance from the square of the original metric. As a consequence, we obtain a refined estimate of the error term in the asymptotic expansion of the volume of (sub-)Riemannian metric balls. To achieve this, we develop a novel technique to perturb sub-Riemannian geodesics, allowing us to modify their endpoints in a prescribed vertical direction.

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