Sard property for rank 2 polarizations in metabelian Lie groups
arXiv:2607.12530
The paper establishes dimension bounds for abnormal and Goh‑abnormal sets in rank‑2 polarized metabelian Lie groups, proving that the endpoint map satisfies the Sard property and that the resulting sub‑Riemannian structures have the minimizing Sard property.
Abstract
We provide bounds on the dimension of the abnormal set for rank 2 polarizations on metabelian Lie groups, establishing the Sard property for the end-point map of such groups. We also obtain bounds for the dimension of the Goh-abnormal set for metabelian Lie groups where the codimension of the derived subgroup is at most 2, with no assumption on the rank of the polarization. We thus infer that these polarized groups, equipped with sub-Riemannian structures, satisfy the minimizing Sard property.
20 pages