On the structural theory of ${\mathrm{II}}_{1}$ factors of negatively curved groups
[Sur la structure des facteurs de type ${\mathrm{II}}_{1}$ associé avec les groupes de courbure négative]
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 $\mathrm{Sp}\left(n,1\right)$, $n\ge 2$, 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 $\mathrm{Sp}\left(n,1\right)$, $n\ge 2$, are virtually ${W}^{*}$-superrigid.

DOI : https://doi.org/10.24033/asens.2183
Classification : 46L10,  20F67
Mots clés : Forte solidité, groupes de courbure négative, groupes «bi-exacts»
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Chifan, Ionut; Sinclair, Thomas. On the structural theory of ${\rm II}_1$ factors of negatively curved groups. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 46 (2013) no. 1, pp. 1-33. doi : 10.24033/asens.2183. http://www.numdam.org/articles/10.24033/asens.2183/

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