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Hypergenerated Carnot groups

Enrico Le Donne, Luca Nalon, Nicola Paddeu, Simone Verzellesi

TL;DR

The paper characterizes hypergenerated Carnot groups algebraically and links this structure to sharp geometric regularity: sets with locally constant normal have vertical hyperplane boundaries precisely when the group is hypergenerated, and on non-characteristic hypersurfaces the intrinsic and restricted distances are locally bi-Lipschitz. The authors develop an algebraic reduction to step $2$ via the Kaplan operator, establish existence results for hypergenerated algebras of arbitrary step and order, and classify low-dimensional indecomposable examples. They further extend these results to higher codimension submanifolds and leverage weak tangents of Lipschitz and contact maps to connect local geometry with global distance properties. The work unifies perimeter regularity, sub-Riemannian geometry, and algebraic structure, with implications for MinRank-type problems in the characterization process and broad applicability to the regularity theory in stratified groups.

Abstract

In this paper we provide an algebraic characterization of those stratified groups in which boundaries with locally constant normal are locally flat. We show that these groups, which we call hypergenerated, are exactly the stratified groups where embeddings of non-characteristic hypersurfaces are locally bi-Lipschitz. Finally, we extend these results to submanifolds of arbitrary codimension.

Hypergenerated Carnot groups

TL;DR

The paper characterizes hypergenerated Carnot groups algebraically and links this structure to sharp geometric regularity: sets with locally constant normal have vertical hyperplane boundaries precisely when the group is hypergenerated, and on non-characteristic hypersurfaces the intrinsic and restricted distances are locally bi-Lipschitz. The authors develop an algebraic reduction to step via the Kaplan operator, establish existence results for hypergenerated algebras of arbitrary step and order, and classify low-dimensional indecomposable examples. They further extend these results to higher codimension submanifolds and leverage weak tangents of Lipschitz and contact maps to connect local geometry with global distance properties. The work unifies perimeter regularity, sub-Riemannian geometry, and algebraic structure, with implications for MinRank-type problems in the characterization process and broad applicability to the regularity theory in stratified groups.

Abstract

In this paper we provide an algebraic characterization of those stratified groups in which boundaries with locally constant normal are locally flat. We show that these groups, which we call hypergenerated, are exactly the stratified groups where embeddings of non-characteristic hypersurfaces are locally bi-Lipschitz. Finally, we extend these results to submanifolds of arbitrary codimension.

Paper Structure

This paper contains 13 sections, 26 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Equiregular sub-Riemannian manifolds
  4. Carnot groups
  5. Embedded surfaces in Carnot groups
  6. Algebraic structure of hypergenerated groups
  7. A characterization of hypergenerated Lie algebras
  8. Hypergenerated groups up to dimension $7$
  9. Existence of hypergenerated Lie algebras of large step
  10. Hypersurfaces with locally constant normal
  11. Non-characteristic surfaces
  12. Weak tangents of Lipschitz maps
  13. Weak tangents of sub-Riemannian manifolds

Key Result

Theorem 1.1

Let $\mathbb{G}$ be a Carnot group. The following are equivalent:

Theorems & Definitions (70)

  • Theorem 1.1
  • Definition 2.1: Stratified basis
  • Definition 2.2
  • Definition 3.1: Hypergenerated algebras and groups
  • Remark 3.2
  • Proposition 3.3
  • proof
  • Corollary 3.4
  • Remark 3.5
  • Remark 3.6
  • ...and 60 more