## Explicit examples of equivalence relations and II$_1$ factors with prescribed fundamental group and outer automorphism group

HTML articles powered by AMS MathViewer

- by Steven Deprez PDF
- Trans. Amer. Math. Soc.
**367**(2015), 6837-6876 Request permission

## Abstract:

In this paper we give a number of explicit constructions for II$_1$ factors and II$_1$ equivalence relations that have prescribed fundamental group and outer automorphism group. We construct factors and relations that have uncountable fundamental group different from $\mathbb {R}_{+}^{\ast }$. In fact, given any II$_1$ equivalence relation, we construct a II$_1$ factor with the same fundamental group. Given any locally compact unimodular second countable group $G$, our construction gives a II$_1$ equivalence relation $\mathcal {R}$ whose outer automorphism group is $G$. The same construction does not give a II$_1$ factor with $G$ as outer automorphism group, but when $G$ is a compact group or if $G=\mathrm {SL}^{\pm }_n\mathbb {R}=\{g\in \mathrm {GL}_n\mathbb {R}\mid \det (g)=\pm 1\}$, then we still find a type II$_1$ factor whose outer automorphism group is $G$.## References

- Jon Aaronson,
*The intrinsic normalising constants of transformations preserving infinite measures*, J. Analyse Math.**49**(1987), 239–270. MR**928513**, DOI 10.1007/BF02792898 - 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 - Bachir Bekka, Pierre de la Harpe, and Alain Valette,
*Kazhdan’s property (T)*, New Mathematical Monographs, vol. 11, Cambridge University Press, Cambridge, 2008. MR**2415834**, DOI 10.1017/CBO9780511542749 - Robert J. Blattner,
*Automorphic group representations*, Pacific J. Math.**8**(1958), 665–677. MR**103421**, DOI 10.2140/pjm.1958.8.665 - Ionut Chifan and Cyril Houdayer,
*Bass-Serre rigidity results in von Neumann algebras*, Duke Math. J.**153**(2010), no. 1, 23–54. MR**2641939**, DOI 10.1215/00127094-2010-020 - A. Connes,
*Classification of injective factors. Cases $II_{1},$ $II_{\infty },$ $III_{\lambda },$ $\lambda \not =1$*, Ann. of Math. (2)**104**(1976), no. 1, 73–115. MR**454659**, DOI 10.2307/1971057 - 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 - J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson,
*$\Bbb {ATLAS}$ of finite groups*, Oxford University Press, Eynsham, 1985. Maximal subgroups and ordinary characters for simple groups; With computational assistance from J. G. Thackray. MR**827219**
[Deprez:PhDthesis] S. Deprez, Some computations of invariants of type II$_1$ factors, PhD thesis, K.U.Leuven, 2011. math.ku.dk/~sdeprez/publications-en.html
- Steven Deprez and Stefaan Vaes,
*A classification of all finite index subfactors for a class of group-measure space $\textrm {II}_1$ factors*, J. Noncommut. Geom.**5**(2011), no. 4, 523–545. MR**2838524**, DOI 10.4171/JNCG/85 - Sébastien Falguières and Stefaan Vaes,
*Every compact group arises as the outer automorphism group of a $\textrm {II}_1$ factor*, J. Funct. Anal.**254**(2008), no. 9, 2317–2328. MR**2409162**, DOI 10.1016/j.jfa.2008.02.002 - 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 - M. Gromov,
*Hyperbolic groups*, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75–263. MR**919829**, DOI 10.1007/978-1-4613-9586-7_{3} - Cyril Houdayer,
*Construction of type $\rm II_1$ factors with prescribed countable fundamental group*, J. Reine Angew. Math.**634**(2009), 169–207. MR**2560409**, DOI 10.1515/CRELLE.2009.072 - Cyril Houdayer, Sorin Popa, and Stefaan Vaes,
*A class of groups for which every action is $\mathrm {W}^*$-superrigid*, Groups Geom. Dyn.**7**(2013), no. 3, 577–590. MR**3095710**, DOI 10.4171/GGD/198 - Cyril Houdayer and Éric Ricard,
*Approximation properties and absence of Cartan subalgebra for free Araki-Woods factors*, Adv. Math.**228**(2011), no. 2, 764–802. MR**2822210**, DOI 10.1016/j.aim.2011.06.010 - 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 - 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 - A. Yu. Ol′shanskiĭ,
*On residualing homomorphisms and $G$-subgroups of hyperbolic groups*, Internat. J. Algebra Comput.**3**(1993), no. 4, 365–409. MR**1250244**, DOI 10.1142/S0218196793000251 - Narutaka Ozawa,
*There is no separable universal $\rm II_1$-factor*, Proc. Amer. Math. Soc.**132**(2004), no. 2, 487–490. MR**2022373**, DOI 10.1090/S0002-9939-03-07127-2 - 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. I*, Invent. Math.**165**(2006), no. 2, 369–408. MR**2231961**, DOI 10.1007/s00222-006-0501-4 - 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,
*Cocycle and orbit equivalence superrigidity for malleable actions of $w$-rigid groups*, Invent. Math.**170**(2007), no. 2, 243–295. MR**2342637**, DOI 10.1007/s00222-007-0063-0 - 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 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 - Sorin Popa and Stefaan Vaes,
*Actions of $\Bbb F_\infty$ whose $\textrm {II}_1$ factors and orbit equivalence relations have prescribed fundamental group*, J. Amer. Math. Soc.**23**(2010), no. 2, 383–403. MR**2601038**, DOI 10.1090/S0894-0347-09-00644-4 - Sorin Popa and Stefaan Vaes,
*Group measure space decomposition of $\textrm {II}_1$ factors and $W^\ast$-superrigidity*, Invent. Math.**182**(2010), no. 2, 371–417. MR**2729271**, DOI 10.1007/s00222-010-0268-5 - Sorin Popa and Stefaan Vaes,
*Cocycle and orbit superrigidity for lattices in $\textrm {SL}(n,\Bbb R)$ acting on homogeneous spaces*, Geometry, rigidity, and group actions, Chicago Lectures in Math., Univ. Chicago Press, Chicago, IL, 2011, pp. 419–451. MR**2807839** - Sorin Popa and Stefaan Vaes,
*On the fundamental group of $\textrm {II}_1$ factors and equivalence relations arising from group actions*, Quanta of maths, Clay Math. Proc., vol. 11, Amer. Math. Soc., Providence, RI, 2010, pp. 519–541. MR**2732063**, DOI 10.1007/s00222-010-0268-5 - I. M. Singer,
*Automorphisms of finite factors*, Amer. J. Math.**77**(1955), 117–133. MR**66567**, DOI 10.2307/2372424 - Stefaan Vaes,
*Rigidity results for Bernoulli actions and their von Neumann algebras (after Sorin Popa)*, Astérisque**311**(2007), Exp. No. 961, viii, 237–294. Séminaire Bourbaki. Vol. 2005/2006. MR**2359046** - 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

## Additional Information

**Steven Deprez**- Affiliation: Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B, B-3001 Leuven, Belgium
- Email: steven.f.l.deprez@gmail.com
- Received by editor(s): October 5, 2012
- Received by editor(s) in revised form: April 9, 2013
- Published electronically: June 18, 2015
- Additional Notes: The author was a research assistant of the Research Foundation – Flanders (FWO) (until August 2011) and a postdoc at the University of Copenhagen (from September 2011). The author was partially supported by ERC Grant VNALG-200749 and ERC Advanced Grant no. OAFPG 247321, and was supported by the Danish National Research Foundation (DNRF) through the Centre for Symmetry and Deformation
- © Copyright 2015 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**367**(2015), 6837-6876 - MSC (2010): Primary 46L36; Secondary 28D15, 46L40, 37A20
- DOI: https://doi.org/10.1090/tran/6298
- MathSciNet review: 3378816