Rigidity for group actions on homogeneous spaces by affine transformations

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, 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 Popa [On a class of type II1 factors with Betti numbers invariants. Ann. of Math. (2) 163(3) (2006), 809-899] if and only if the induced action of Gamma on P(g) admits no Gamma-invariant probability measure, where g is the Lie algebra of G. This generalizes results of Burger [Kazhdan constants for SL (3, 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.