[Paper Review] Stability of compact actions of the Heisenberg group

Tania M. Begazo, Nicolau C. Saldanha|ArXiv.org|Dec 27, 2004

Advanced Differential Equations and Dynamical Systems7 references3 citations

TL;DR

This paper investigates the $C^1$ stability of compact actions of the Heisenberg group on 4-manifolds, focusing on homogeneous horizontal actions. It establishes that the stability of orbits—defined via $C^1$ perturbations of vector fields satisfying the Heisenberg commutator relations—depends critically on the derivative of the structure matrix $A^\sharp(0)$: nilpotency implies $L$-instability, non-vanishing determinant implies $L$-stability, and absence of real eigenvalues implies $T$-stability, demonstrating non-equivalence of stability notions in the nilpotent setting.

ABSTRACT

Let G be the Heisenberg group of real lower triangular 3x3 matrices with unit diagonal. A locally free smooth action of G on a manifold M^4 is given by linearly independent vector fields X_1, X_2, X_3 such that X_3 = [X_1,X_2] and [X_1,X_3] = [X_2, X_3] = 0. The C^1 topology for vector fields induces a topology in the space of actions of G on M^4. An action is compact if all orbits are compact. Given a compact action $θ$, we investigate under which conditions its C^1 perturbations $ ildeθ$ are guaranteed to be compact. There is more than one interesting definition of stability, and we show that in the case of the Heisenberg group, unlike for actions of R^n, the definitions do not turn out to be equivalent.

Motivation & Objective

• To determine under which conditions $C^1$-small perturbations of a compact action of the Heisenberg group on a 4-manifold remain compact.
• To clarify the distinction between two stability notions—$L$-stability (local instability) and $T$-stability (total stability)—in the context of nilpotent group actions.
• To analyze homogeneous horizontal actions on $M = (G/H) \times (-\epsilon, \epsilon)$, where $G$ is the Heisenberg group and $H$ is a discrete cocompact subgroup.
• To characterize the stability of the orbit $(G/H) \times \{0\}$ in terms of the matrix $A^\sharp(0)$, the derivative of the structure matrix $A(z)$ at $z=0$.
• To prove that for the Heisenberg group, unlike for $\mathbb{R}^n$, the notions of $L$-stability and $T$-stability are not equivalent.

Proposed method

• Model the action via a smooth family of automorphisms $\phi_z$ of the Heisenberg group $G$, inducing a homogeneous horizontal action on $M = (G/H) \times (-\epsilon, \epsilon)$.
• Represent the homomorphism $\phi_z$ by a matrix $A(z)$ with entries satisfying $a_{13}(z) = a_{23}(z) = 0$ and $a_{33}(z) = \det A^\sharp(z)$, where $A^\sharp(z)$ is the $2\times 2$ top-left block.
• Define $C^1$-stability using perturbations of the generating vector fields $X_1, X_2, X_3$ satisfying the Heisenberg commutator relations $[X_1,X_2] = X_3$, $[X_1,X_3] = [X_2,X_3] = 0$.
• Introduce the abelian holonomy map $\tau_{\operatorname{Ab},p}(v)$, measuring the asymptotic drift of holonomy along orbits, and use its constancy or variation across orbits to detect non-compactness.
• Use the integral of $\operatorname{Ab}'(\tilde{\tau}_p(v))$ along vertical lines to detect non-vanishing drift, leading to contradiction if perturbations preserve compactness.
• Apply unique ergodicity of a certain vector field (Section 4) and topological arguments on foliations to analyze the behavior of orbits under perturbation.

Experimental results

Research questions

• RQ1 Under what conditions on the derivative $A^\sharp(0)$ of the structure matrix is the orbit $(G/H) \times \{0\}$ $L$-stable under $C^1$ perturbations of the action?
• RQ2 When is the orbit $T$-stable, meaning all nearby $C^1$-perturbations of the vector fields yield compact orbits?
• RQ3 Why do the notions of $L$-stability and $T$-stability fail to be equivalent for the Heisenberg group, unlike in the case of $\mathbb{R}^n$ actions?
• RQ4 How does the presence of real eigenvalues or nilpotency in $A^\sharp(0)$ affect the stability of the orbit?
• RQ5 What role does the abelian holonomy map $\tau_{\operatorname{Ab},p}(v)$ play in detecting non-compactness of perturbed orbits?

Key findings

• If $A^\sharp(0)$ is nilpotent, the orbit is $L$-unstable: arbitrarily small $C^1$-perturbations can produce non-compact orbits.
• If $A^\sharp(0)$ has non-zero determinant, the orbit is $L$-stable: all sufficiently small $C^1$-perturbations yield compact orbits.
• If $A^\sharp(0)$ has no real eigenvalues, the orbit is $T$-stable: all $C^1$-perturbations of the vector fields yield compact orbits.
• If $A^\sharp(0) = \operatorname{diag}(-\lambda, 2\lambda, \lambda)$ with $\lambda \neq 0$, the orbit is $L$-stable but $T$-unstable, showing non-equivalence of stability concepts.
• The abelian holonomy map $\tau_{\operatorname{Ab},p}(v)$ is constant on connected components of the complement of compact orbits, and its variation along vertical lines detects non-compactness.
• The proof relies on showing that a certain integral of the derivative of the holonomy map cannot vanish if $A^\sharp(z)$ is invertible and non-degenerate, contradicting the existence of a non-compact orbit.