Actions with globally hypoelliptic leafwise Laplacian and rigidity

TL;DR

This paper investigates smooth actions of b^k with globally hypoelliptic leafwise Laplacian, establishing their ergodic and cohomological properties, and demonstrating local rigidity for specific b^2 actions on nilmanifolds under Diophantine conditions.

Contribution

It proves that such actions are uniquely ergodic, often have finite-dimensional cohomology, and shows local rigidity for certain b^2 actions on nilmanifolds.

Findings

Actions are necessarily uniquely ergodic and minimal.

Cohomology groups are often finite-dimensional or trivial.

Certain b^2 actions on nilmanifolds are locally rigid under Diophantine conditions.

Abstract

We prove several results concerning smooth $\mathbb R^k$ actions with the property that their leafwise Laplacian is globally hypoelliptic. Such actions are necessarily uniquely ergodic and minimal, and cohomology is often finite-dimensional, even trivial. Further we consider a class of examples of $\mathbb R^2$ actions on 2-step nilmanifolds, which have globally hypoelliptic leafwise Laplacian, and we show transversal local rigidity under certain Diophantine conditions.