Strong rigidity of \(\text{II}_1\) factors arising from malleable actions of \(w\)-rigid groups. II
The paper expands its author's programme on von Neumann algebra rigidity, which has recently led to spectacular results on the fundamental group of II\(_1\) factors [Ann. Math. (2) 163, 809--899 (2006; Zbl 1120.46045), part I of the paper, Invent. Math. 165, 369--408 (2006; Zbl 1120.46043). Isomorphisms between group measure space II\(_1\) factors \(L^\infty (X,\mu) \rtimes_\sigma G\) arising from free ergodic measure-preserving actions of discrete countable groups on probability spaces are considered this time. The typical assumption on the group \(G\) is to be either \textit{ICC} (infinite conjugacy class) and \(w\)-rigid (i.e., to contain an infinite normal subgroup \(H\) with relative Kazhdan-Margulis property T), or to belong to the class \(w{\mathcal T}_0\) of groups with infinite subgroups \(H\) with relative property T and such that \(H\) is weakly quasi normal in \(G\) and is not virtually Abelian. The typical assumption on the action \(\sigma\) is to be sub \(s\)-malleable (as defined in part I and clustering (a strong notion of mixing defined in the paper). Bernoulli actions of ICC groups \(G\), acting on infinite product probability spaces \((X,\mu)=\prod_{g\in G} (X_0,\mu_0)\) by \(\sigma_{g_0} ((x_g)_g)=(x_{g_0^{-1} g})_g\), provide examples of sub \(s\)-malleable clustering actions. Measure group II\(_1\) factors are shown to exhibit very strong rigidity properties under these kinds of assumptions, in the sense that any isomorphism between \(M_1 =L^\infty (X_1,\mu_1)\rtimes_{\sigma_1} G_1\) and \(L^\infty (X_2,\mu_2) \rtimes_{\sigma_2} G_2\) is essentially implemented by a probability space isomorphism \((X_1,\mu_1)\cong (X_2,\mu_2)\) which conjugates the actions \(\sigma_1\) and \(\sigma_2\) with respect to a group isomorphism \(G_1 \cong G_2\). In fact, the following more general result is proved. Theorem. Let \(G_i\) be a countable discrete group acting freely and ergodically by \(\sigma_i\) on a standard probability space \((X_i,\mu_i)\) and denote \(M_i=L^\infty (X_i,\mu_i) \rtimes_{\sigma_i} G_i\), \(i=1,2\). Assume that \(\theta :M_1 \cong pM_2 p\) is an isomorphism for some projection \(p\in M_2\). Assume also that \(G_1\) is either \(w\)-rigid and ICC or \(G_1\in w{\mathcal T}_0\), and that \(\sigma_2\) is sub \(s\)-malleable and clustering. Then \(p=1\) and there exist a character group isomorphism \(\delta :G_1 \cong G_2\), a probability space isomorphism \(\Delta :(X_1,\mu_1) \cong (X_2,\mu_2)\) with \(\sigma_2 (\delta (g))\Delta=\Delta \sigma_1 (g)\) for all \(g\in G_1\), a unitary \(u\in M_2\) such that \(\theta=\text{ Ad}(u)\circ \theta^\gamma \circ \theta^{\delta,\Delta}\), where \(\sigma^\gamma\) denotes the natural automorphism of \(M_2\) implemented by \(\gamma\) and \(\theta^{\delta,\Delta}:M_1 \cong M_2\) the natural isomorphism implemented by \(\delta\) and \(\Delta\). A corollary of the previous theorem shows that, given two \(w\)-rigid ICC groups \(G_1\) and \(G_2\), the II\(_1\) factors of the wreath products \(\tilde{G}_i={\mathbb Z}^{G_i} \rtimes G_i\), \(i=1,2\), are isomorphic iff \(\tilde{G}_1 \cong \tilde{G}_2\) and iff \(G_1 \cong G_2\). This result can be regarded as a group measure space relative version of Connes' longstanding conjecture that II\(_1\) factors arising from ICC groups with property T are isomorphic iff the groups are isomorphic [cf., e.g., \textit{A. Connes}, Noncommutative Geometry, Academic Press (1994; Zbl 0818.46076)]. Another corollary is relevant for the computation of the outer automorphism group of a factor \(M=L^\infty (X,\mu)\rtimes_\sigma G\), shown to coincide with the commutant of \(\sigma\) in the automorphisms of the probability space \((X,\mu)\) whenever \(\sigma\) is a Bernoulli \(G\)-action of an ICC group \(G\) which is either \(w\)-rigid or belongs to the class \(w{\mathcal T}_0\). The crux of this work is a technical criterion for conjugacy of actions (Theorem 5.2) which extends results from part I. Its proof is intricate and occupies the first six sections of the paper. It is noticed that this result also holds in the case of amenable groups, thus giving an affirmative answer to a conjecture of \textit{S. Neshveyev} and \textit{E. Størmer} [J. Funct. Anal. 195, 239--261 (2002; Zbl 1022.46041)] when their weakly mixing condition is replaced by the stronger assumption of clustering. The final section provides some supplementary applications to OE (orbit equivalence) rigidity, in the framework of measurable equivalence relations. These results are related with previous work on OE of \textit{D. Gaboriau} [Publ. Math. IHES 95, 93--150 (2002; Zbl 1022.37002)], \textit{A. Furman} [Ann. Math. (2) 150, 1059--1081 (1999; Zbl 0943.22013), 1083--1108 (1999; Zbl 0943.22012)], and with recent work of \textit{N. Monod} and \textit{Y. Shalom} on OE superrigidity [Ann. Math. (2) 164, 825--878 (2006)].
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