Actions of 𝔽_{∞} whose II₁ factors and orbit equivalence relations have prescribed fundamental group

Popa, Sorin ¹   ; Vaes, Stefaan ²

¹ Department of Mathematics, University of California at Los Angeles, Los Angeles, California 90095-1555
² Department of Mathematics, K.U.Leuven, Celestijnenlaan 200B, B–3001 Leuven, Belgium

Journal of the American Mathematical Society, Tome 23 (2010) no. 2, pp. 383-403

Résumé

We show that given any subgroup $\mathcal {F}$ of $\mathbb {R}_+$ which is either countable or belongs to a certain “large” class of uncountable subgroups, there exist continuously many free ergodic measure-preserving actions $\sigma _i$ of the free group with infinitely many generators $\mathbb {F}_\infty$ on probability measure spaces $(X_i,\mu _i)$ such that their associated group measure space II$_1$ factors $M_i=\operatorname {L}^\infty (X_i) \rtimes _{\sigma _i} \mathbb {F}_\infty$ and orbit equivalence relations $\mathcal {R}_i=\mathcal {R} (\mathbb {F}_\infty {\overset {}{\curvearrowright }} X_i)$ have fundamental group equal to $\mathcal {F}$ and with $M_i$ (respectively $\mathcal {R}_i$) stably non-isomorphic. Moreover, these actions can be taken so that $\mathcal {R}_i$ has no outer automorphisms and any automorphism of $M_i$ is unitarily conjugate to an automorphism that acts trivially on the subalgebra $\operatorname {L}^\infty (X_i)$ of $M_i$.

Détail

DOI : 10.1090/S0894-0347-09-00644-4

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