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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Actions of $\mathbb {F}_\infty$ whose II$_1$ factors and orbit equivalence relations have prescribed fundamental group
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by Sorin Popa and Stefaan Vaes PDF
J. Amer. Math. Soc. 23 (2010), 383-403 Request permission

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
  • MR Author ID: 141080
  • Email: popa@math.ucla.edu
  • Stefaan Vaes
  • Affiliation: Department of Mathematics, K.U.Leuven, Celestijnenlaan 200B, B–3001 Leuven, Belgium
  • Email: stefaan.vaes@wis.kuleuven.be
  • 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.
  • © Copyright 2009 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: J. Amer. Math. Soc. 23 (2010), 383-403
  • MSC (2000): Primary 46L10; Secondary 37A20, 28D15
  • DOI: https://doi.org/10.1090/S0894-0347-09-00644-4
  • MathSciNet review: 2601038