On the structural theory of ${\mathrm{II}}_1$ factors of negatively curved groups

Chifan, Ionut; Sinclair, Thomas

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 46 (2013) no. 1, pp. 1-33.

Ozawa a montré dans [21] que, pour un groupe c.c.i. hyperbolique, le facteur de type ${\mathrm{II}}_1$ 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 $\mathrm{Sp}(n,1)$, $n\ge 2$, sont virtuellement ${\mathrm{W}}^{*}$-superrigides.

Ozawa showed in [21] that for any i.c.c. hyperbolic group, the associated group factor $L\Gamma$ 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 $L\Gamma$ is strongly solid. Using our methods in cooperation with a cocycle superrigidity result of Ioana [12], we show that profinite actions of lattices in $\mathrm{Sp}(n,1)$, $n\ge 2$, are virtually $W^{*}$-superrigid.

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