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

[Sur la structure des facteurs de type $\rm II_1$ associé avec les groupes de courbure négative]

Chifan, Ionut ; Sinclair, Thomas

Annales scientifiques de l'École Normale Supérieure, Tome 46 (2013), p. 1-33 / Harvested from Numdam

Résumé

Ozawa a montré dans [21] que, pour un groupe c.c.i. hyperbolique, le facteur de type $\rm II_1$ 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 ${\rm Sp}(n,1)$, $n\ge 2$, sont virtuellement $\rm W^*$-superrigides.

Abstract

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 ${\rm Sp}(n,1)$, $n\ge 2$, are virtually $W^*$-superrigid.

Détail

Publié le : 2013-01-01

MR 3087388 | Zbl 1290.46053

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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