Rigidity for group actions on homogeneous spaces by affine transformations

TL;DR

This paper establishes a criterion for the rigidity of group actions on homogeneous spaces, linking it to the absence of invariant measures on a projective Lie algebra space, and applies it to actions on nilmanifolds.

Contribution

It provides a new criterion for rigidity of actions on homogeneous spaces based on invariant measures, generalizing previous results and applying to nilmanifold automorphisms.

Findings

Rigidity characterized by absence of invariant measures on projective Lie algebra

Generalizes previous rigidity results by Burger, Ioana, and Shalom

Establishes rigidity for automorphism actions on certain nilmanifolds

Abstract

We give a criterion for the rigidity of actions on homogeneous spaces. Let $G$ be a real Lie group, $\Lambda$ a lattice in $G$, and $\Gamma$ a subgroup of the affine group Aff$(G)$ stabilizing $\Lambda$. Then the action of $\Gamma$ on $G/\Lambda$ has the rigidity property in the sense of S. Popa, if and only if the induced action of $\Gamma$ on $\mathbb{P}(\frak{g})$ admits no $\Gamma$-invariant probability measure, where $\frak{g}$ is the Lie algebra of $G$. This generalizes results of M. Burger, and A. Ioana and Y. Shalom. As an application, we establish rigidity for the action of a class of groups acting by automorphisms on nilmanifolds associated to step 2 nilpotent Lie groups.