Semigenerated step-3 Carnot algebras and applications to sub-Riemannian perimeter

TL;DR

This paper characterizes semigenerated Carnot groups, especially step-3 groups, and explores their implications for the structure of sets with finite intrinsic perimeter, introducing new algebraic classes and rectifiability results.

Contribution

It provides a complete algebraic characterization of semigeneration in step-3 Carnot groups and introduces new classes of groups to study perimeter properties.

Findings

Semigeneration occurs iff the semigroup generated by horizontal half-spaces is a vertical half-space.

Step-3 Carnot groups without Engel-type quotients are semigenerated.

In certain classes, sets of finite perimeter are strongly rectifiable.

Abstract

This paper contributes to the study of sets of finite intrinsic perimeter in Carnot groups. Our intent is to characterize in which groups the only sets with constant intrinsic normal are the vertical half-spaces. Our viewpoint is algebraic: such a phenomenon happens if and only if the semigroup generated by each horizontal half-space is a vertical half-space. We call \emph{semigenerated} those Carnot groups with this property. For Carnot groups of nilpotency step 3 we provide a complete characterization of semigeneration in terms of whether such groups do not have any Engel-type quotients. Engel-type groups, which are introduced here, are the minimal (in terms of quotients) counterexamples. In addition, we give some sufficient criteria for semigeneration of Carnot groups of arbitrary step. For doing this, we define a new class of Carnot groups, which we call type $(\Diamond)$ and which generalizes the previous notion of type $(\star)$ defined by M. Marchi. As an application, we get that in type $ (\Diamond) $ groups and in step 3 groups that do not have any Engel-type algebra as a quotient, one achieves a strong rectifiability result for sets of finite perimeter in the sense of Franchi, Serapioni, and Serra-Cassano.