Rigidity for group actions on homogeneous spaces by affine transformations

Abstract

We give a criterion for the rigidity of the action of a group of affine transformations of a homogeneous space of a real Lie group. Let $G$ be a real Lie group, $\unicode[STIX]{x1D6EC}$ a lattice in $G$, and $\unicode[STIX]{x1D6E4}$ a subgroup of the affine group $\text{Aff}(G)$ stabilizing $\unicode[STIX]{x1D6EC}$. Then the action of $\unicode[STIX]{x1D6E4}$ on $G/\unicode[STIX]{x1D6EC}$ has the rigidity property in the sense of Popa [On a class of type $\text{II}_{1}$ factors with Betti numbers invariants. Ann. of Math. (2)163(3) (2006), 809–899] if and only if the induced action of $\unicode[STIX]{x1D6E4}$ on $\mathbb{P}(\mathfrak{g})$ admits no $\unicode[STIX]{x1D6E4}$-invariant probability measure, where $\mathfrak{g}$ is the Lie algebra of $G$. This generalizes results of Burger [Kazhdan constants for $\text{SL}(3,\mathbf{Z})$. J. Reine Angew. Math.413 (1991), 36–67] and Ioana and Shalom [Rigidity for equivalence relations on homogeneous spaces. Groups Geom. Dyn.7(2) (2013), 403–417]. As an application, we establish rigidity for the action of a class of groups acting by automorphisms on nilmanifolds associated to two-step nilpotent Lie groups.