## 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
- J. Amer. Math. Soc.
**23**(2010), 383-403 - DOI: https://doi.org/10.1090/S0894-0347-09-00644-4
- Published electronically: August 26, 2009
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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 $M_i$.## References

- Jon Aaronson and Mahendra Nadkarni,
*$L_\infty$ eigenvalues and $L_2$ spectra of nonsingular transformations*, Proc. London Math. Soc. (3)**55**(1987), no. 3, 538–570. MR**907232**, DOI 10.1112/plms/s3-55.3.538 - M. Burger,
*Kazhdan constants for $\textrm {SL}(3,\textbf {Z})$*, J. Reine Angew. Math.**413**(1991), 36–67. MR**1089795**, DOI 10.1515/crll.1991.413.36 - Alain Connes,
*Une classification des facteurs de type $\textrm {III}$*, Ann. Sci. École Norm. Sup. (4)**6**(1973), 133–252 (French). MR**341115** - A. Connes,
*A factor of type $\textrm {II}_{1}$ with countable fundamental group*, J. Operator Theory**4**(1980), no. 1, 151–153. MR**587372** - A. Connes and V. Jones,
*A $\textrm {II}_{1}$ factor with two nonconjugate Cartan subalgebras*, Bull. Amer. Math. Soc. (N.S.)**6**(1982), no. 2, 211–212. MR**640947**, DOI 10.1090/S0273-0979-1982-14981-3 - Jacob Feldman and Calvin C. Moore,
*Ergodic equivalence relations, cohomology, and von Neumann algebras. I*, Trans. Amer. Math. Soc.**234**(1977), no. 2, 289–324. MR**578656**, DOI 10.1090/S0002-9947-1977-0578656-4 - Alex Furman,
*Orbit equivalence rigidity*, Ann. of Math. (2)**150**(1999), no. 3, 1083–1108. MR**1740985**, DOI 10.2307/121063 - Alex Furman,
*Outer automorphism groups of some ergodic equivalence relations*, Comment. Math. Helv.**80**(2005), no. 1, 157–196. MR**2130572**, DOI 10.4171/CMH/10 - Damien Gaboriau,
*Invariants $l^2$ de relations d’équivalence et de groupes*, Publ. Math. Inst. Hautes Études Sci.**95**(2002), 93–150 (French). MR**1953191**, DOI 10.1007/s102400200002 - Damien Gaboriau,
*Coût des relations d’équivalence et des groupes*, Invent. Math.**139**(2000), no. 1, 41–98 (French, with English summary). MR**1728876**, DOI 10.1007/s002229900019 - Damien Gaboriau and Sorin Popa,
*An uncountable family of nonorbit equivalent actions of $\Bbb F_n$*, J. Amer. Math. Soc.**18**(2005), no. 3, 547–559. MR**2138136**, DOI 10.1090/S0894-0347-05-00480-7 - Sergey L. Gefter,
*Outer automorphism group of the ergodic equivalence relation generated by translations of dense subgroup of compact group on its homogeneous space*, Publ. Res. Inst. Math. Sci.**32**(1996), no. 3, 517–538. MR**1409801**, DOI 10.2977/prims/1195162855 - Sergey L. Gefter and Valentin Ya. Golodets,
*Fundamental groups for ergodic actions and actions with unit fundamental groups*, Publ. Res. Inst. Math. Sci.**24**(1988), no. 6, 821–847 (1989). MR**1000122**, DOI 10.2977/prims/1195173929 - T. Giordano and G. Skandalis,
*Krieger factors isomorphic to their tensor square and pure point spectrum flows*, J. Funct. Anal.**64**(1985), no. 2, 209–226. MR**812392**, DOI 10.1016/0022-1236(85)90075-8 - Greg Hjorth,
*A lemma for cost attained*, Ann. Pure Appl. Logic**143**(2006), no. 1-3, 87–102. MR**2258624**, DOI 10.1016/j.apal.2005.05.034 - Bernard Host, Jean-François Méla, and François Parreau,
*Nonsingular transformations and spectral analysis of measures*, Bull. Soc. Math. France**119**(1991), no. 1, 33–90 (English, with French summary). MR**1101939** - Adrian Ioana, Jesse Peterson, and Sorin Popa,
*Amalgamated free products of weakly rigid factors and calculation of their symmetry groups*, Acta Math.**200**(2008), no. 1, 85–153. MR**2386109**, DOI 10.1007/s11511-008-0024-5 - Jean-Pierre Kahane and Raphaël Salem,
*Ensembles parfaits et séries trigonométriques*, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1301, Hermann, Paris, 1963 (French). MR**0160065** - Alexander S. Kechris,
*Classical descriptive set theory*, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR**1321597**, DOI 10.1007/978-1-4612-4190-4 - Barthélemy Le Gac,
*Some properties of Borel subgroups of real numbers*, Proc. Amer. Math. Soc.**87**(1983), no. 4, 677–680. MR**687640**, DOI 10.1090/S0002-9939-1983-0687640-X - V. Mandrekar, M. Nadkarni, and D. Patil,
*Singular invariant measures on the line*, Studia Math.**35**(1970), 1–13. MR**259069**, DOI 10.4064/sm-35-1-1-13 - Nicolas Monod and Yehuda Shalom,
*Orbit equivalence rigidity and bounded cohomology*, Ann. of Math. (2)**164**(2006), no. 3, 825–878. MR**2259246**, DOI 10.4007/annals.2006.164.825 - F. J. Murray and J. Von Neumann,
*On rings of operators*, Ann. of Math. (2)**37**(1936), no. 1, 116–229. MR**1503275**, DOI 10.2307/1968693 - F. J. Murray and J. von Neumann,
*On rings of operators. IV*, Ann. of Math. (2)**44**(1943), 716–808. MR**9096**, DOI 10.2307/1969107 - Remus Nicoara, Sorin Popa, and Roman Sasyk,
*On $\textrm {II}_1$ factors arising from 2-cocycles of $w$-rigid groups*, J. Funct. Anal.**242**(2007), no. 1, 230–246. MR**2274021**, DOI 10.1016/j.jfa.2006.05.015 - N. Ozawa and S. Popa, On a class of $\mathrm {II}_1$ factors with at most one Cartan subalgebra.
*Ann. of Math.*(2), to appear. arXiv:0706.3623 - S. Popa, Correspondences, INCREST preprint No. 56/1986, www.math.ucla.edu/~popa/ preprints.html.
- Sorin Popa,
*On a class of type $\textrm {II}_1$ factors with Betti numbers invariants*, Ann. of Math. (2)**163**(2006), no. 3, 809–899. MR**2215135**, DOI 10.4007/annals.2006.163.809 - Sorin Popa,
*Strong rigidity of $\rm II_1$ factors arising from malleable actions of $w$-rigid groups. II*, Invent. Math.**165**(2006), no. 2, 409–451. MR**2231962**, DOI 10.1007/s00222-006-0502-3 - Sorin Popa,
*On the superrigidity of malleable actions with spectral gap*, J. Amer. Math. Soc.**21**(2008), no. 4, 981–1000. MR**2425177**, DOI 10.1090/S0894-0347-07-00578-4 - Sorin Popa,
*Some computations of 1-cohomology groups and construction of non-orbit-equivalent actions*, J. Inst. Math. Jussieu**5**(2006), no. 2, 309–332. MR**2225044**, DOI 10.1017/S1474748006000016 - Sorin Popa and Dimitri Shlyakhtenko,
*Universal properties of $L(\textbf {F}_\infty )$ in subfactor theory*, Acta Math.**191**(2003), no. 2, 225–257. MR**2051399**, DOI 10.1007/BF02392965 - Sorin Popa and Stefaan Vaes,
*Strong rigidity of generalized Bernoulli actions and computations of their symmetry groups*, Adv. Math.**217**(2008), no. 2, 833–872. MR**2370283**, DOI 10.1016/j.aim.2007.09.006 - S. Popa and S. Vaes, On the fundamental group of II$_1$ factors and equivalence relations arising from group actions. To appear in
*Noncommutative geometry, Proceedings of the Conference in honor of A.Connes’ 60th birthday.*arXiv:0810.0706 - I. M. Singer,
*Automorphisms of finite factors*, Amer. J. Math.**77**(1955), 117–133. MR**66567**, DOI 10.2307/2372424 - Asger Törnquist,
*Orbit equivalence and actions of $\Bbb F_n$*, J. Symbolic Logic**71**(2006), no. 1, 265–282. MR**2210067**, DOI 10.2178/jsl/1140641174 - Stefaan Vaes,
*Explicit computations of all finite index bimodules for a family of $\textrm {II}_1$ factors*, Ann. Sci. Éc. Norm. Supér. (4)**41**(2008), no. 5, 743–788 (English, with English and French summaries). MR**2504433**, DOI 10.24033/asens.2081

## Bibliographic 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