Parameter rigid actions of simply connected nilpotent Lie groups

TL;DR

This paper establishes conditions under which actions of simply connected nilpotent Lie groups on compact manifolds are parameter rigid, generalizing previous results from Heisenberg groups to broader classes of nilpotent Lie groups.

Contribution

It proves a criterion linking cocycle cohomology to parameter rigidity and constructs new examples of such actions for Lie groups with rational graded Lie algebras.

Findings

Parameter rigidity characterized by cocycle cohomology conditions.

Construction of new parameter rigid actions for Lie groups with rational structures.

Generalization of previous results from Heisenberg groups to broader nilpotent Lie groups.

Abstract

We show that for a locally free action of a simply connected nilpotent Lie group on a compact manifold, if every real valued cocycle is cohomologous to a constant cocycle, then the action is parameter rigid. The converse is true if the action has a dense orbit. Using this, we construct parameter rigid actions of simply connected nilpotent Lie groups whose Lie algebras admit rational structures with graduations. This generalizes the results of dos Santos [Parameter rigid actions of the Heisenberg groups. Ergod. Th. & Dynam. Sys. 27 (2007), 1719-1735] concerning the Heisenberg groups.