On relative property (T) and Haagerup’s property

Chifan, Ionut; Ioana, Adrian

doi:10.1090/S0002-9947-2011-05259-1

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Browser Compatibility Info Help
ISSN 1088-6850(online) ISSN 0002-9947(print)

On relative property (T) and Haagerup's property

Authors: Ionut Chifan and Adrian Ioana
Journal: Trans. Amer. Math. Soc. 363 (2011), 6407-6420
MSC (2010): Primary 20F69; Secondary 46L10
Posted: July 14, 2011
MathSciNet review: 2833560
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the following three properties for countable discrete groups $\Gamma$ : (1) $\Gamma$ has an infinite subgroup with relative property (T), (2) the group von Neumann algebra $L\Gamma$ has a diffuse von Neumann subalgebra with relative property (T) and (3) $\Gamma$ does not have Haagerup's property. It is clear that (1) $\Longrightarrow$ (2) $\Longrightarrow$ (3). We prove that both of the converses are false.

[Bu91] M. Burger, Kazhdan constants for 𝑆𝐿(3,𝑍), J. Reine Angew. Math. 413 (1991), 36–67. MR 1089795 (92c:22013), http://dx.doi.org/10.1515/crll.1991.413.36
[Ch83] Marie Choda, Group factors of the Haagerup type, Proc. Japan Acad. Ser. A Math. Sci. 59 (1983), no. 5, 174–177. MR 718798 (85f:46117)
[CJ85] A. Connes and V. Jones, Property 𝑇 for von Neumann algebras, Bull. London Math. Soc. 17 (1985), no. 1, 57–62. MR 766450 (86a:46083), http://dx.doi.org/10.1112/blms/17.1.57
[CCJJV01] Pierre-Alain Cherix, Michael Cowling, Paul Jolissaint, Pierre Julg, and Alain Valette, Groups with the Haagerup property, Progress in Mathematics, vol. 197, Birkhäuser Verlag, Basel, 2001. Gromov’s a-T-menability. MR 1852148 (2002h:22007)
[Co06a] Yves de Cornulier, Kazhdan and Haagerup properties in algebraic groups over local fields, J. Lie Theory 16 (2006), no. 1, 67–82. MR 2196414 (2006i:22018)
[Co06b] Yves de Cornulier, Relative Kazhdan property, Ann. Sci. École Norm. Sup. (4) 39 (2006), no. 2, 301–333 (English, with English and French summaries). MR 2245534 (2007c:22008), http://dx.doi.org/10.1016/j.ansens.2005.12.006
[CSV09] Y. de Cornulier, Y. Stalder, A. Valette: Proper actions of wreath products and generalizations, preprint arXiv:0905.3960.
[Fu09] A. Furman: A survey of Measured Group Theory, preprint arXiv:0901.0678.
[GHW05] Erik Guentner, Nigel Higson, and Shmuel Weinberger, The Novikov conjecture for linear groups, Publ. Math. Inst. Hautes Études Sci. 101 (2005), 243–268. MR 2217050 (2007c:19007), http://dx.doi.org/10.1007/s10240-005-0030-5
[Io07] Adrian Ioana, Rigidity results for wreath product 𝐼𝐼₁ factors, J. Funct. Anal. 252 (2007), no. 2, 763–791. MR 2360936 (2008j:46046), http://dx.doi.org/10.1016/j.jfa.2007.04.005
[Io09] Adrian Ioana, Relative property (T) for the subequivalence relations induced by the action of 𝑆𝐿₂(ℤ) on 𝕋², Adv. Math. 224 (2010), no. 4, 1589–1617. MR 2646305 (2011f:22008), http://dx.doi.org/10.1016/j.aim.2010.01.021
[Ka67] D. A. Každan, On the connection of the dual space of a group with the structure of its closed subgroups, Funkcional. Anal. i Priložen. 1 (1967), 71–74 (Russian). MR 0209390 (35 #288)
[Ma82] G. A. Margulis, Finitely-additive invariant measures on Euclidean spaces, Ergodic Theory Dynam. Systems 2 (1982), no. 3-4, 383–396 (1983). MR 721730 (85g:28004), http://dx.doi.org/10.1017/S014338570000167X
[Ma91] G. A. Margulis, Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 17, Springer-Verlag, Berlin, 1991. MR 1090825 (92h:22021)
[MvN36] F. J. Murray and J. Von Neumann, On rings of operators, Ann. of Math. (2) 37 (1936), no. 1, 116–229. MR 1503275, http://dx.doi.org/10.2307/1968693
[Pe09] J. Peterson: Examples of group actions which are virtually W*-superrigid, preprint 2009.
[Po06a] Sorin Popa, On a class of type 𝐼𝐼₁ factors with Betti numbers invariants, Ann. of Math. (2) 163 (2006), no. 3, 809–899. MR 2215135 (2006k:46097), http://dx.doi.org/10.4007/annals.2006.163.809
[Po06b] Sorin Popa, Strong rigidity of 𝐼𝐼₁ factors arising from malleable actions of 𝑤-rigid groups. I, Invent. Math. 165 (2006), no. 2, 369–408. MR 2231961 (2007f:46058), http://dx.doi.org/10.1007/s00222-006-0501-4
[Po07] Sorin Popa, Deformation and rigidity for group actions and von Neumann algebras, International Congress of Mathematicians. Vol. I, Eur. Math. Soc., Zürich, 2007, pp. 445–477. MR 2334200 (2008k:46186), http://dx.doi.org/10.4171/022-1/18
[Wi08] D. Witte Morris: Introduction to Arithmetic Groups, lecture notes, available at http://people.uleth.ca/ dave.morris/.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 20F69, 46L10

Retrieve articles in all journals with MSC (2010): 20F69, 46L10

Additional Information

Ionut Chifan
Affiliation: Department of Mathematics, 1326 Stevenson Center, Vanderbilt University, Nash- ville, Tennessee 37240 – and – Institute of Mathematics of the Romanian Academy, Bucharest, Romania
Email: ionut.chifan@vanderbilt.edu

Adrian Ioana
Affiliation: Department of Mathematics, University of California, Los Angeles, Los Angeles, California 90095-155505 – and – Institute of Mathematics of the Romanian Academy, Bucharest, Romania
Email: adiioana@math.ucla.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05259-1
PII: S 0002-9947(2011)05259-1
Received by editor(s): July 14, 2009
Received by editor(s) in revised form: November 23, 2009
Posted: July 14, 2011
Additional Notes: The second author was supported by a Clay Research Fellowship
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.