Free products, orbit equivalence and measure equivalence rigidity

Abstract

We study the analogue in orbit equivalence of free product decomposition and free indecomposability for countable groups. We introduce the (orbit equivalence invariant) notion of freely indecomposable (FI) standard probability measure preserving equivalence relations and establish a criterion to check it, namely non-hyperfiniteness and vanishing of the first L 2 -Betti number. We obtain Bass-Serre rigidity results, i.e. forms of uniqueness in free product decompositions of equivalence relations with (FI) components. The main features of our work are weak algebraic assumptions and no ergodicity hypothesis for the components. We deduce, for instance, that a measure equivalence between two free products of non-amenable groups with vanishing first ℓ 2 -Betti numbers is induced by measure equivalences of the components. We also deduce new classification results in Orbit Equivalence and II 1 factors.

Some OE- and W⁎-rigidity results for actions by wreath product groups

“…Further he showed that, given two stably orbit equivalent actions, σ and ρ, of such groups with σ | Γ and ρ| Λ ergodic, one has σ | Γ and ρ| Λ are stably orbit equivalent. He was also able to prove a similar measure equivalence rigidity for certain classes of direct products and amalgamated free products, thus obtaining rigidity results á la Monod and Shalom [20], as well as of Bass-Serre type [16,1,2]. His methods rely on Ozawa's techniques [23,24] involving the class S of groups, being C * -algebraic in nature and depending crucially on exactness of the groups involved.…”

“…So let us assume that θ t does not converge uniformly on (P ) 1 . Therefore by transversality of θ t , Theorem 1.1, one can find a constant 0 < c < 1, and infinite sequences t k ∈ R, u k ∈ U (P ) such that t k → 0 and θ t k (u k ) − E M θ t k (u k ) 2 c.…”

“…Since Q has property (T), they must converge uniformly on the unit ball of Q. Thus there is a finite set F ⊂ H I such that if x = g∈H I x g u g ∈ (Q) 1 then g∈F x g u g 2 > 1 2 for all x ∈ (Q) 1 . This implies that g∈F E A (xu * g ) 2 > 1 2 and by Theorem 2.1 we obtain Q ≺ M A, which is a contradiction because A is abelian while Q has property (T).…”

“…First we use the spectral gap argument to show that the deformation θ converges to the identity uniformly on (Q) 1 . Indeed, exactly as in [32], since P has no amenable direct summand, Lemma 4.2 implies that P ∩M ω ⊂ M ω .…”

“…So now we are left to deal with the case when θ t does converge uniformly on (P ) 1 . In this case Theorem 3.1 implies that P ≺ M A Γ or P ≺ M A H F for some finite set F ⊂ Γ .…”