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Actions of $ \mathbb{F}_\infty$ whose II$ _1$ factors and orbit equivalence relations have prescribed fundamental group

Authors: Sorin Popa and Stefaan Vaes
Journal: J. Amer. Math. Soc. 23 (2010), 383-403
MSC (2000): Primary 46L10; Secondary 37A20, 28D15
Published electronically: August 26, 2009
MathSciNet review: 2601038
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Abstract: 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$.

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Additional Information

Sorin Popa
Affiliation: Department of Mathematics, University of California at Los Angeles, Los Angeles, California 90095-1555

Stefaan Vaes
Affiliation: Department of Mathematics, K.U.Leuven, Celestijnenlaan 200B, B–3001 Leuven, Belgium

Keywords: Fundamental group of II$_1$ factors, fundamental group of II$_1$ equivalence relations, outer automorphism group, actions of free groups, rigid actions, deformation/rigidity.
Received by editor(s): June 3, 2008
Published electronically: August 26, 2009
Additional Notes: The first author was partially supported by NSF Grant DMS-0601082
The second author was partially supported by Research Programme G.0231.07 of the Research Foundation—Flanders (FWO) and the Marie Curie Research Training Network Non-Commutative Geometry MRTN-CT-2006-031962. The second author would like to thank the Department of Mathematics at UCLA for their warm hospitality during the work on this paper.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.