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Classification of abelian actions with globally hypoelliptic orbitwise laplacian I: The Greenfield-Wallach conjecture on nilmanifolds

Sven Sandfeldt

TL;DR

The paper classifies globally hypoelliptic (GH) abelian ${ m R}^{k}$-actions on closed manifolds via the orbitwise Laplacian $oldsymbol{ riangle}_{ ext{α}}=- extstyleig(X_{1}^{2}+ frac{}{k}X_{k}^{2}ig)$, proving that GH actions on compact nilmanifolds are smoothly conjugate to translations on nilmanifolds in three regimes. It establishes a finite-dimensional cohomology for GH actions, with $H^{ullet}(oldsymbol{ ext{α}})\congoldsymbol{igwedge}^{ullet}(oldsymbol{ ext{R}}^{k})$, and develops a cocycle-rigidity framework for nilpotent targets to obtain translation models in the nilmanifold case. The results extend the Greenfield–Wallach conjecture to nilmanifolds and, under topological or codimension constraints (large $b_{1}$ or codimension one), show that GH actions must be translations on nilmanifolds or closely related structures such as Heisenberg-type bases. The work provides tools for cohomological analysis and tame estimates that underpin local rigidity and yields explicit obstructions on which nilmanifolds can support GH actions. Altogether, this advances understanding of dynamical rigidity for higher-rank abelian actions in the GH regime and highlights when GH dynamics reduce to linear translations on nilmanifolds.

Abstract

For a $\mathbb{R}^{k}-$action generated by vector fields $X_{1},...,X_{k}$ we define an operator $-(X_{1}^{2}+...+X_{k}^{2})$, the orbitwise laplacian. In this paper, we study and classify $\mathbb{R}^{k}-$actions whose orbitwise laplacian is globally hypoelliptic (GH). In three different settings we prove that any such action is given by a translation action on some compact nilmanifold, (i) when the space is a compact nilmanifold, (ii) when the first Betti number of the manifold is sufficiently large, (iii) when the codimension of the orbitfoliation of the action is $1$. As a consequence, we prove the Greenfield-Wallach conjecture on all nilmanifolds. Along the way, we also calculate the cohomology of GH $\mathbb{R}^{k}-$actions, proving, in particular, that it is always finite dimensional.

Classification of abelian actions with globally hypoelliptic orbitwise laplacian I: The Greenfield-Wallach conjecture on nilmanifolds

TL;DR

The paper classifies globally hypoelliptic (GH) abelian -actions on closed manifolds via the orbitwise Laplacian , proving that GH actions on compact nilmanifolds are smoothly conjugate to translations on nilmanifolds in three regimes. It establishes a finite-dimensional cohomology for GH actions, with , and develops a cocycle-rigidity framework for nilpotent targets to obtain translation models in the nilmanifold case. The results extend the Greenfield–Wallach conjecture to nilmanifolds and, under topological or codimension constraints (large or codimension one), show that GH actions must be translations on nilmanifolds or closely related structures such as Heisenberg-type bases. The work provides tools for cohomological analysis and tame estimates that underpin local rigidity and yields explicit obstructions on which nilmanifolds can support GH actions. Altogether, this advances understanding of dynamical rigidity for higher-rank abelian actions in the GH regime and highlights when GH dynamics reduce to linear translations on nilmanifolds.

Abstract

For a action generated by vector fields we define an operator , the orbitwise laplacian. In this paper, we study and classify actions whose orbitwise laplacian is globally hypoelliptic (GH). In three different settings we prove that any such action is given by a translation action on some compact nilmanifold, (i) when the space is a compact nilmanifold, (ii) when the first Betti number of the manifold is sufficiently large, (iii) when the codimension of the orbitfoliation of the action is . As a consequence, we prove the Greenfield-Wallach conjecture on all nilmanifolds. Along the way, we also calculate the cohomology of GH actions, proving, in particular, that it is always finite dimensional.
Paper Structure (16 sections, 26 theorems, 123 equations)

This paper contains 16 sections, 26 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. Background and definitions
  3. Cohomology of dynamical systems
  4. Globally hypoelliptic operators and tame estimates
  5. The orbitwise laplacian
  6. Nilpotent Lie groups
  7. Properties of the orbitwise laplacian
  8. Orbitwise laplacian and cohomology
  9. Betti number and invariant 1-forms
  10. Tame estimates
  11. Rigidity of globally hypoelliptic actions
  12. Proof of Theorem \ref{['MainThm:ThmB']} (i): The Greenfield-Wallach conjecture on nilmanifolds
  13. Proof of Theorem \ref{['MainThm:ThmB']} (ii): Manifolds with large first Betti number
  14. Proof of Theorem \ref{['MainThm:ThmB']} point (iii): Actions with small codimension
  15. A nilmanifold without globally hypoelliptic action
  16. ...and 1 more sections

Key Result

Theorem A

The Greenfield-Wallach conjecture is true on all compact nilmanifolds. That is, the only nilmanifold that supports a globally hypoelliptic vector field is the torus.

Theorems & Definitions (71)

  • Theorem A
  • Theorem B
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Definition 2.1
  • Example 2.1
  • Definition 2.2
  • Theorem 2.1
  • ...and 61 more