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, Λ a lattice in G, and Γ a subgroup of the affine group Aff(G) stabilizing Λ. Then the action of Γ on G/Λ has the rigidity property in the sense of S. Popa [Pop06], if and only if the induced action of Γ on P(g) admits no Γ-invariant probability measure, where g is the Lie algebra of G. This generalizes results of M. Burger [Bur91], and A. Ioana and Y. Shalom [IS13]. 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.

Existence of rigid actions for finitely-generated non-amenable linear groups

“…The crucial tool for the proof of our results is an extension of our characterization of rigid actions on homogeneous spaces of real Lie groups (Theorem 1.3, [Bou15]) to the framework of p-adic groups, where p ∈ P, the set of prime numbers.…”

Cocycle and orbit equivalence superrigidity for coinduced actions

“…See also [Io09] for an ergodic theoretic characterization of rigidity. See [IS10, Theorem D] for a more general statement regarding Example 1.4(ii) (see also [Bo15] for other examples). In addition to these concrete classes of rigid actions, note that any free group Γ = Γ 1 * Γ 2 , with |Γ 1 | ≥ 2 and |Γ 2 | ≥ 3, admits uncountably many non-orbit equivalent free ergodic p.m.p.…”