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Explicit examples of equivalence relations and II\(_{1}\) factors with prescribed fundamental group and outer automorphism group. (English) Zbl 1344.46044

Summary: 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.

MSC:

46L36 Classification of factors
28D15 General groups of measure-preserving transformations
46L40 Automorphisms of selfadjoint operator algebras
37A20 Algebraic ergodic theory, cocycles, orbit equivalence, ergodic equivalence relations
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[1] Aaronson, Jon, The intrinsic normalising constants of transformations preserving infinite measures, J. Analyse Math., 49, 239-270 (1987) · Zbl 0644.28013 · doi:10.1007/BF02792898
[2] Aaronson, Jon; Nadkarni, Mahendra, \(L_\infty\) eigenvalues and \(L_2\) spectra of nonsingular transformations, Proc. London Math. Soc. (3), 55, 3, 538-570 (1987) · Zbl 0636.28010 · doi:10.1112/plms/s3-55.3.538
[3] [BekkaDelaharpeValette] B. Bekka, P. de la Harpe, and A. Valette. \newblockKazhdan’s Property (T), \newblock Cambridge University Press, 2008. · Zbl 1146.22009
[4] Blattner, Robert J., Automorphic group representations, Pacific J. Math., 8, 665-677 (1958) · Zbl 0087.32001
[5] Chifan, Ionut; Houdayer, Cyril, Bass-Serre rigidity results in von Neumann algebras, Duke Math. J., 153, 1, 23-54 (2010) · Zbl 1201.46057 · doi:10.1215/00127094-2010-020
[6] Connes, A., Classification of injective factors. Cases \(II_1, II_{\infty }, III_{\lambda }, \lambda \not =1\), Ann. of Math. (2), 104, 1, 73-115 (1976)
[7] Connes, A., A factor of type \({\rm II}_1\) with countable fundamental group, J. Operator Theory, 4, 1, 151-153 (1980) · Zbl 0455.46056
[8] Connes, A.; Jones, V., A \({\rm II}_1\) factor with two nonconjugate Cartan subalgebras, Bull. Amer. Math. Soc. (N.S.), 6, 2, 211-212 (1982) · Zbl 0501.46056 · doi:10.1090/S0273-0979-1982-14981-3
[9] Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; Wilson, R. A., Atlas of finite groups, xxxiv+252 pp. (1985), Oxford University Press: Eynsham:Oxford University Press
[10] [Deprez:PhDthesis] S. Deprez, \newblockSome computations of invariants of type \(II_1\) factors, \newblock PhD thesis, K.U.Leuven, 2011. \newblockmath.ku.dk/\textasciitildesdeprez/publications-en.html
[11] Deprez, Steven; Vaes, Stefaan, A classification of all finite index subfactors for a class of group-measure space \({\rm II}_1\) factors, J. Noncommut. Geom., 5, 4, 523-545 (2011) · Zbl 1235.46059 · doi:10.4171/JNCG/85
[12] Falgui{\`“e}res, S{\'”e}bastien; Vaes, Stefaan, Every compact group arises as the outer automorphism group of a \({\rm II}_1\) factor, J. Funct. Anal., 254, 9, 2317-2328 (2008) · Zbl 1153.46036 · doi:10.1016/j.jfa.2008.02.002
[13] Feldman, Jacob; Moore, Calvin C., Ergodic equivalence relations, cohomology, and von Neumann algebras. I, Trans. Amer. Math. Soc., 234, 2, 289-324 (1977) · Zbl 0369.22009
[14] Gromov, M., Hyperbolic groups. Essays in group theory, Math. Sci. Res. Inst. Publ. 8, 75-263 (1987), Springer: New York:Springer · doi:10.1007/978-1-4613-9586-7\_3
[15] Houdayer, Cyril, Construction of type \(\rm II_1\) factors with prescribed countable fundamental group, J. Reine Angew. Math., 634, 169-207 (2009) · Zbl 1209.46038 · doi:10.1515/CRELLE.2009.072
[16] Houdayer, Cyril; Popa, Sorin; Vaes, Stefaan, A class of groups for which every action is \(\text{W}^*\)-superrigid, Groups Geom. Dyn., 7, 3, 577-590 (2013) · Zbl 1314.46072 · doi:10.4171/GGD/198
[17] Houdayer, Cyril; Ricard, {\'E}ric, Approximation properties and absence of Cartan subalgebra for free Araki-Woods factors, Adv. Math., 228, 2, 764-802 (2011) · Zbl 1267.46071 · doi:10.1016/j.aim.2011.06.010
[18] Ioana, Adrian; Peterson, Jesse; Popa, Sorin, Amalgamated free products of weakly rigid factors and calculation of their symmetry groups, Acta Math., 200, 1, 85-153 (2008) · Zbl 1149.46047 · doi:10.1007/s11511-008-0024-5
[19] Murray, F. J.; von Neumann, J., On rings of operators. IV, Ann. of Math. (2), 44, 716-808 (1943) · Zbl 0060.26903
[20] Ol{\cprime }shanski{\u \i }, A. Yu., On residualing homomorphisms and \(G\)-subgroups of hyperbolic groups, Internat. J. Algebra Comput., 3, 4, 365-409 (1993) · Zbl 0830.20053 · doi:10.1142/S0218196793000251
[21] Ozawa, Narutaka, There is no separable universal \(\rm II_1\)-factor, Proc. Amer. Math. Soc., 132, 2, 487-490 (electronic) (2004) · Zbl 1041.46045 · doi:10.1090/S0002-9939-03-07127-2
[22] Popa, Sorin, On a class of type \({\rm II}_1\) factors with Betti numbers invariants, Ann. of Math. (2), 163, 3, 809-899 (2006) · Zbl 1120.46045 · doi:10.4007/annals.2006.163.809
[23] Popa, Sorin, Strong rigidity of \(\rm II_1\) factors arising from malleable actions of \(w\)-rigid groups. I, Invent. Math., 165, 2, 369-408 (2006) · Zbl 1120.46043 · doi:10.1007/s00222-006-0501-4
[24] Popa, Sorin, Strong rigidity of \(\rm II_1\) factors arising from malleable actions of \(w\)-rigid groups. II, Invent. Math., 165, 2, 409-451 (2006) · Zbl 1120.46044 · doi:10.1007/s00222-006-0502-3
[25] Popa, Sorin, Cocycle and orbit equivalence superrigidity for malleable actions of \(w\)-rigid groups, Invent. Math., 170, 2, 243-295 (2007) · Zbl 1131.46040 · doi:10.1007/s00222-007-0063-0
[26] Popa, Sorin, On the superrigidity of malleable actions with spectral gap, J. Amer. Math. Soc., 21, 4, 981-1000 (2008) · Zbl 1222.46048 · doi:10.1090/S0894-0347-07-00578-4
[27] Popa, Sorin; Vaes, Stefaan, Strong rigidity of generalized Bernoulli actions and computations of their symmetry groups, Adv. Math., 217, 2, 833-872 (2008) · Zbl 1137.37003 · doi:10.1016/j.aim.2007.09.006
[28] Popa, Sorin; Vaes, Stefaan, Actions of \(\mathbb{F}_\infty\) whose \({\rm II}_1\) factors and orbit equivalence relations have prescribed fundamental group, J. Amer. Math. Soc., 23, 2, 383-403 (2010) · Zbl 1202.46069 · doi:10.1090/S0894-0347-09-00644-4
[29] Popa, Sorin; Vaes, Stefaan, Group measure space decomposition of \({\rm II}_1\) factors and \(W^\ast \)-superrigidity, Invent. Math., 182, 2, 371-417 (2010) · Zbl 1238.46052 · doi:10.1007/s00222-010-0268-5
[30] Popa, Sorin; Vaes, Stefaan, Cocycle and orbit superrigidity for lattices in \({\rm SL}(n,\mathbb{R})\) acting on homogeneous spaces. Geometry, rigidity, and group actions, Chicago Lectures in Math., 419-451 (2011), Univ. Chicago Press: Chicago, IL:Univ. Chicago Press · Zbl 1291.37006
[31] Popa, Sorin; Vaes, Stefaan, On the fundamental group of \({\rm II}_1\) factors and equivalence relations arising from group actions. Quanta of maths, Clay Math. Proc. 11, 519-541 (2010), Amer. Math. Soc.: Providence, RI:Amer. Math. Soc. · Zbl 1222.37008
[32] Singer, I. M., Automorphisms of finite factors, Amer. J. Math., 77, 117-133 (1955) · Zbl 0064.11001
[33] Vaes, Stefaan, Rigidity results for Bernoulli actions and their von Neumann algebras (after Sorin Popa), Ast\'erisque, 311, Exp. No. 961, viii, 237-294 (2007) · Zbl 1194.46085
[34] Vaes, Stefaan, Explicit computations of all finite index bimodules for a family of \({\rm II}_1\) factors, Ann. Sci. \'Ec. Norm. Sup\'er. (4), 41, 5, 743-788 (2008) · Zbl 1194.46086
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