On globally hypoelliptic abelian actions and their existence on homogeneous spaces

TL;DR

This paper explores the existence and classification of globally hypoelliptic abelian actions on homogeneous spaces, conjecturing their rarity outside nilmanifolds and providing partial non-existence results for certain cases.

Contribution

It conjectures that globally hypoelliptic actions on homogeneous spaces only occur on nilmanifolds and proves non-existence for specific actions on certain homogeneous spaces.

Findings

Non-existence of globally hypoelliptic $R^2$ actions with quasi-unipotent generators on $SL(n, R)$ homogeneous spaces.

Smooth conjugacy of such actions on solvmanifolds to actions on nilmanifolds.

Partial support for the conjecture regarding the rarity of globally hypoelliptic actions outside nilmanifolds.

Abstract

We define globally hypoelliptic smooth $\mathbb R^k$ actions as actions whose leafwise Laplacian along the orbit foliation is a globally hypoelliptic differential operator. When $k=1$, strong global rigidity is conjectured for such actions by Greenfield-Wallach and Katok: every such action is smoothly conjugate to a Diophantine flow on the torus. The conjecture has been confirmed for all homogeneous flows on homogeneous spaces \cite{FFRH}. In this paper we conjecture that among homogeneous $\mathbb R^k$ actions ($k\ge 2$) on homogeneous spaces globally hypoelliptic actions exist only on nilmanifolds. We obtain a partial result towards this conjecture: we show non-existence of globally hypoelliptic $\mathbb R^2$ actions on homogeneous spaces $G/\Gamma$, with at least one quasi-unipotent generator, where $G= SL(n, \mathbb R)$. We also show that the same type of actions on solvmanifolds are smoothly conjugate to homogeneous actions on nilmanifolds.