Rigidity for equivalence relations on homogeneous spaces

Abstract

Abstract. We study Popa's notion of rigidity for equivalence relations induced by actions on homogeneous spaces. For any lattices and ƒ in a semisimple Lie group G with finite center and no compact factors we prove that the action Õ G=ƒ is rigid. If in addition G has property (T) then we derive that the von Neumann algebra L 1 .G=ƒ/ Ì has property (T). We also show that if the stabilizer of any non-zero point in the Lie algebra of G under the adjoint action of G is amenable (e.g., if G D SL 2 .R/), then any ergodic subequivalence relation of the orbit equivalence relation of the action Õ G=ƒ is either hyperfinite or rigid.Mathematics Subject Classification (2010). 37A20, 46L36.

Rigidity for group actions on homogeneous spaces by affine transformations

“…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.…”

“…We should mention that our approach is largely based on techniques developed in [IS13]. Up until recently (see [IS13]), the examples of actions of group which have the property (T) relative to the space were built using the relative property (T) of a pair (A ⋊ Γ, A).…”

Strong ergodicity, property (T), and orbit equivalence rigidity for translation actions

“…Then Examples 1.7-1.9 below provide many examples of countable dense subgroups Γ < G such that the translation action Γ G is strongly ergodic. On the other hand, by [IS10,Theorem D], the action Γ G/Σ is rigid, in the sense of S. Popa [Po01]. Altogether, this shows that Theorem E applies to a large family of rigid actions.…”

Classification and rigidity for von Neumann algebras

“…T 2 [Po01b] and SL 2 (Z) SL 2 (R)/SL 2 (Z) [IS10]. By [Ga08] any free product group Γ = Γ 1 * Γ 2 with |Γ 1 | 2 and |Γ 2 | 3 admits a continuum of rigid actions whose II 1 factors are mutually non-isomorphic.…”

“…A von Neumann subalgebra P of a tracial von Neumann algebra (M, τ ) has the relative property (T) if any deformation φ n : M → M of the identity of M must converge uniformly to the identity on the unital ball of P [Po01b].Note that if P = M , then this property amounts to the property (T) of M , in the sense Connes and Jones[CJ83].Given two countable groups Γ 0 < Γ, the inclusion of group von Neumann algebras L(Γ 0 ) ⊂ L(Γ) has the relative property (T) if and only if the inclusion Γ 0 < Γ has the relative property (T)[Po01b]. Examples of inclusions of groups with the relative property (T) include SL n (Z) < SL n (Z), for n 3 [Ka67], and Z 2 < Z 2 ⋊ Γ, for any non-amenable subgroup Γ < SL 2 (Z)[Bu91].Several classes of inclusions of von Neumann algebras inclusions satisfying the relative property (T) that do not arise from inclusions of groups have been recently found in [Io09,CI09,IS10]. For instance, the main result of[Io09] asserts that if M ⊂ L(Z 2 ⋊ SL 2 (Z)) is a non-hyperfinite subfactor which contains L(Z 2 ), then the inclusion L(Z 2 ) ⊂ M has the relative property (T).If (M, τ ) is a tracial von Neumann algebra, then an M -M bimodule is a Hilbert space endowed with commuting * -representations of M and its opposite algebra, M op .…”