Strong rigidity of II1 factors arising from malleable actions of w-rigid groups, II

Abstract

Abstract. We prove that any isomorphism θ : M 0 ≃ M of group measure space II 1 factors, M 0 = L ∞ (X 0 , µ 0 ) ⋊ σ 0 G 0 , M = L ∞ (X, µ) ⋊ σ G, with G 0 an ICC group containing an infinite normal subgroup with the relative property (T) of KazhdanMargulis (i.e. G 0 w-rigid) and σ a Bernoulli shift action of some group G, essentially comes from an isomorphism of probability spaces which conjugates the actions with respect to some identification G 0 ≃ G. Moreover, any isomorphism θ of M 0 onto a "corner" pMp of M, for p ∈ M an idempotent, forces p = 1. In particular, all group measure space factors associated with Bernoulli shift actions of w-rigid ICC groups have trivial fundamental group and any isomorphism of such factors comes from an isomorphism of the corresponding groups. This settles a "group measure space version" of Connes rigidity conjecture, shown in fact to hold true in a greater generality than just for ICC property (T) groups. We apply these results to ergodic theory, establishing new strong rigidity and superrigidity results for orbit equivalence relations. Introduction.We continue in this paper the study of rigidity properties of isomorphisms θ : M 0 ≃ M of crossed product II 1 factors initiated in ([Po4]), concentrating here on the "group measure space" case, when the factors M 0 , M involved come from free ergodic measure preserving (m.p.) actions of groups on probability spaces. Similarly to ([Po4]), we assume the "source" factor M 0 comes from an arbitrary free ergodic measure preserving action σ 0 of a "mildly rigid" group G 0 (i.e. having a "large" subgroup with the relative property (T) of Kazhdan-Margulis), while the "target" factor M comes from an action σ satisfying good "deformation+mixing"

Strong rigidity of II1 factors arising from malleable actions of w-rigid groups, I

“…Such strong rigidity results will be obtained in the sequel papers with the same title [Po6] (the "group measure space" case) and [Po7] (the "non-classical" case).…”

“…Theorems 4.1, 5.2, 5.3, 5.3 ′ are key ingredients in the proof of strong rigidity results for isomorphisms of cross product factors N 0 ⋊ σ 0 G 0 , N ⋊ σ G for G 0 w-rigid or in the class wT 0 and σ commutative or non-commutative Bernoulli shifts, in ([Po6]). This will allow us to classify large classes of factors N ⋊ σ G, with explicit calculations of various invariants, such as F (N ⋊ σ G).…”

“…Besides its rôle in ( [Po6,7]), this result already enables us to calculate here the fundamental group F (M ) of crossed product II 1 factors M coming from (nonclassical) Connes-Størmer Bernoulli shift actions of arithmetic groups such as G = Z 2 ⋊ Γ, with Γ ⊂ SL(2, Z) a subgroup of finite index, by using results from ([Po3], [Ga]). Thus, if {t i } i are the weights of σ (see [CSt]), then F (M ) is equal to the multiplicative group generated by the ratios {t i /t j } i,j .…”

Stable orbit equivalence of Bernoulli shifts over free groups

“…By contrast, it was shown in [Bo10a] that the main result of [Bo10a] combined with rigidity results of S. Popa [Po06], [Po08] and Y. Kida [Ki08] proves that for many nonamenable groups G, Bernoulli shifts are classified up to orbit equivalence and even stable orbit equivalence by base-space entropy. For example, this includes PSL n .Z/ for n > 2, mapping class groups of surfaces (with a few exceptions) and any nonamenable sofic Ornstein group of the form G D H N with both H and N countably infinite that has no nontrivial finite normal subgroups.…”

A classification of all finite index subfactors for a class of group-measure space $\mathrm{II}_1$ factors

“…This has been successfully applied to the fundamental group (see e.g. [Po01], [Po03], [PV08a]) and the outer automorphism group (see e.g. [IPP05], [PV06], [Va07], [FV07]).…”