Stability of compact actions of the Heisenberg group

TL;DR

This paper investigates the stability of compact actions of the Heisenberg group on 4-manifolds under C^1 perturbations, revealing that different stability definitions are not equivalent in this context.

Contribution

It provides a detailed analysis of stability conditions for Heisenberg group actions, highlighting differences from abelian group actions.

Findings

Not all C^1 perturbations preserve compactness.

Different stability definitions are not equivalent for Heisenberg group actions.

Conditions for stability depend on the specific definition used.

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 $\theta$, we investigate under which conditions its C^1 perturbations $\tilde\theta$ 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.