The Sard problem in step 2 and in filiform Carnot groups

TL;DR

This paper investigates the Sard problem in Carnot groups, providing new bounds for the abnormal set in step 2 groups and characterizing the abnormal set in filiform groups, advancing understanding of geometric control theory.

Contribution

It offers the first lower bounds on the codimension of the abnormal set in step 2 Carnot groups and characterizes the abnormal set in filiform groups as either a line or a variety.

Findings

Lower bounds on the abnormal set codimension in step 2 Carnot groups

Improved bounds over previous results

Characterization of abnormal sets in filiform groups

Abstract

We study the Sard problem for the endpoint map in some well-known classes of Carnot groups. Our first main result deals with step 2 Carnot groups, where we provide lower bounds (depending only on the algebra of the group) on the codimension of the abnormal set; it turns out that our bound is always at least 3, which improves the result proved in arXiv:1503.03610 and settles a question emerged in arXiv:1709.02854. In our second main result we characterize the abnormal set in filiform groups and show that it is either a horizontal line, or a 3-dimensional algebraic variety.