ON THE STRUCTURAL THEORY OF II 1 FACTORS OF NEGATIVELY CURVED GROUPS ʙʏ Iɴ CHIFAN ɴ Tʜ SINCLAIR Aʙʀ.-Ozawa showed in [21] that for any i.c.c. hyperbolic group, the associated group factor LΓ is solid. Developing a new approach that combines some methods of Peterson [29], Ozawa and Popa [27, 28], and Ozawa [25], we strengthen this result by showing that LΓ is strongly solid. Using our methods in cooperation with a cocycle superrigidity result of Ioana [12], we show that profinite actions of lattices in Sp(n, 1), n ≥ 2, are virtually W *-superrigid. R.-Ozawa a montré dans [21] que, pour un groupe c.c.i. hyperbolique, le facteur de type II1 associé est solide. En devéloppant une nouvelle approche, qui combine les méthodes de Peterson [29], d'Ozawa et Popa [27, 28], et d'Ozawa [25], nous renforçons ce résultat en montrant que ce facteur est fortement solide. En combinant nos méthodes avec un résultat d'Ioana de superrigidité des cocycles [12], nous prouvons que les actions des réseaux de Sp(n, 1), n ≥ 2, sont virtuellement W *-superrigides.

Let Γ be a countable group and denote by S the equivalence relation induced by the Bernoulli action Γ [0, 1] Γ , where [0, 1] Γ is endowed with the product Lebesgue measure. We prove that, for any subequivalence relation R of S, there exists a partition {X i } i≥0 of [0, 1] Γ into R-invariant measurable sets such that R |X0 is hyperfinite and R |Xi is strongly ergodic (hence ergodic and non-hyperfinite), for every i ≥ 1.

A. We show that if Γ = Γ 1 × · · · × Γn is a product of n ≥ 2 non-elementary ICC hyperbolic groups then any discrete group Λ which is W * -equivalent to Γ decomposes as a k-fold direct sum exactly when k = n. This gives a group-level strengthening of Ozawa and Popa's unique prime decomposition theorem by removing all assumptions on the group Λ. This result in combination with Margulis' normal subgroup theorem allows us to give examples of lattices in the same Lie group which do not generate stably equivalent II 1 factors.

We obtain new Bass-Serre type rigidity results for II1 equivalence relations and their von Neumann algebras, coming from free ergodic actions of free products of groups on the standard probability space. As an application, we show that any non-amenable factor arising as an amalgamated free product of von Neumann algebras M1 * B M2 over an abelian von Neumann algebra B, is prime, i.e. cannot be written as a tensor product of diffuse factors. This gives, both in the type II1 and in the type III case, new examples of prime factors.2000 Mathematics Subject Classification. 46L10; 46L54; 46L55.