Explicit computations of all finite index bimodules for a family of II${}_1$ factors

Vaes, Stefaan

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 41 (2008) no. 5, pp. 743-788.

Nous étudions des facteurs $M$ et $N$ de type II${}_1$ associés à de bonnes actions Bernoulli généralisées de groupes $\Gamma$ et $\Lambda$ ayant un sous-groupe infini presque-distingué avec la propriété (T) relative. Nous démontrons le résultat de rigidité suivant  : chaque $M$-$N$-bimodule d’indice fini (en particulier, chaque isomorphisme entre $M$ et $N$) peut être décrit par une commensurabilité des groupes $\Gamma$, $\Lambda$ et une commensurabilité de leurs actions. L’algèbre de fusion des $M$-$M$-bimodules d’indice fini est identifiée avec une algèbre de Hecke étendue, ce qui fournit les premiers calculs explicites de l’algèbre de fusion d’un facteur de type II${}_1$. Nous obtenons en particulier des exemples explicites de facteurs II${}_1$ dont l’algèbre de fusion est triviale, ce qui veut dire que tous leurs sous-facteurs d’indice fini sont triviaux.

We study II${}_1$ factors $M$ and $N$ associated with good generalized Bernoulli actions of groups having an infinite almost normal subgroup with the relative property (T). We prove the following rigidity result : every finite index $M$-$N$-bimodule (in particular, every isomorphism between $M$ and $N$) is described by a commensurability of the groups involved and a commensurability of their actions. The fusion algebra of finite index $M$-$M$-bimodules is identified with an extended Hecke fusion algebra, providing the first explicit computations of the fusion algebra of a II${}_1$ factor. We obtain in particular explicit examples of II${}_1$ factors with trivial fusion algebra, i.e. only having trivial finite index subfactors.

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