Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Proper actions of wreath products and generalizations

Authors: Yves Cornulier, Yves Stalder and Alain Valette
Journal: Trans. Amer. Math. Soc. 364 (2012), 3159-3184
MSC (2010): Primary 20F69; Secondary 20E22, 43A05, 43A65
Published electronically: February 9, 2012
MathSciNet review: 2888241
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study stability properties of the Haagerup Property and of coarse embeddability in a Hilbert space, under certain semidirect products. In particular, we prove that they are stable under taking standard wreath products. Our construction also provides a characterization of subsets with relative Property T in a standard wreath product.

References [Enhancements On Off] (What's this?)

  • [AD] G. Arzhantseva, T. Delzant. Examples of random groups, Preprint (2008), available on, to appear in J. of Topol.
  • [AGS] G. Arzhantseva, V. Guba, M. Sapir. Metrics on diagram groups and uniform embeddings in a Hilbert space. Comment. Math. Helv. 81 (2006), no. 4, 911-929. MR 2271228 (2007k:20084)
  • [ANP] T. Austin, A. Naor, Y. Peres. The wreath product of $ \mathbf {Z}$ with $ \mathbf {Z}$ has Hilbert compression exponent 2/3. Proc. Amer. Math. Soc. 137 (2009), 85-90. MR 2439428 (2009f:20060)
  • [BHV] B. Bekka, P. de la Harpe, A. Valette. Kazhdan's Property (T). New Math. Monographs 11, Cambridge Univ. Press, Cambridge 2008. MR 2415834 (2009i:22001)
  • [BDK] J. Bretagnolle, D. Dacunha Castelle, J.-L. Krivine. Lois stables et espaces $ L^p$. Annales de l'Inst. H. Poincaré, section B, tome 2 (1965/1966), 231-259. MR 0203757 (34:3605)
  • [BJS] M. Bożejko, T. Januszkiewicz, R. J. Spatzier. Infinite Coxeter groups do not have Kazhdan's property. J. Operator Theory 19 (1988), no. 1, 63-67. MR 950825 (89i:22025)
  • [Bk] N. Bourbaki. Éléments de Mathématique. Groupes et algèbres de Lie. Chap. 4-6. Hermann, 1968. MR 0453824 (56:12077)
  • [Bo] J. Bourgain. The metrical interpretation of superreflexivity in Banach spaces. Israel J. Math. 56 (1986), no. 2, 222-230. MR 880292 (88e:46007)
  • [BrO] N. Brown, N. Ozawa. C*-algebras and finite-dimensional approximations, Grad. Studies in Math. 88, Amer. Math. Soc., Providence, RI, 2008. MR 2391387 (2009h:46101)
  • [Ch] Guido's book of conjectures (collected by I. Chatterji), Monographie no. 40 de l'Enseign. Math., Genève, 2008. MR 2499538 (2010d:00003)
  • [CDH] I. Chatterji, C. Drutu, F. Haglund. Kazhdan and Haagerup properties from the median viewpoint. Adv. Math. 225(2) (2010), 882-921. MR 2671183 (2011g:20059)
  • [CCJJV] P.-A. Cherix, M. Cowling, P. Jolissaint, P. Julg, A.Valette. Groups with the Haagerup property, Progress in Mathematics, vol. 197. Birkhäuser Verlag, Basel, 2001. MR 1852148 (2002h:22007)
  • [CI] I. Chifan, A. Ioana. On Relative Property (T) and Haagerup's Property. Preprint, arXiv:0906.5363v1, to appear in Trans. Amer. Math. Soc.
  • [CMV] P.-A. Cherix, F. Martin, A. Valette. Spaces with measured walls, the Haagerup property and property (T). Ergodic Theory Dynam. Systems 24 (2004), no. 6, 1895-1908. MR 2106770 (2005i:22006)
  • [Co1] Y. Cornulier. On Haagerup and Kazhdan properties. Thèse EPFL, no 3438 (2006). Dir.: Peter Buser, A. Valette.
  • [Co2] Y. Cornulier. Relative Kazhdan Property. Annales Sci. École Normale Sup. 39 (2006), no. 2, 301-333. MR 2245534 (2007c:22008)
  • [CSV] Y. Cornulier, Y. Stalder, A. Valette. Proper actions of lamplighter groups associated with free groups. C. R. Acad. Sci. Paris, Ser. I 346 (2008), no. 3-4, 173-176. MR 2393636 (2008k:20099)
  • [CT] Y. Cornulier, R. Tessera. Quasi-isometrically embedded free sub-semigroups. Geom. Topol. 12 (2008), 461-473. MR 2390351 (2009c:20072)
  • [CTV] Y. Cornulier, R. Tessera, A. Valette. Isometric group actions on Banach spaces and representations vanishing at infinity. Trans. Groups, 13 (2008), 125-147. MR 2421319 (2009h:22003)
  • [DG] M. Dadarlat, E. Guentner. Constructions preserving Hilbert space uniform embeddability of discrete groups. Trans. Amer. Math. Soc. 355 (2003), no. 8, 3253-3275. MR 1974686 (2004e:20070)
  • [DL] M. Deza, M. Laurent. Geometry of Cuts and Metrics, Springer, Berlin, 1997. MR 1460488 (98g:52001)
  • [Gr] M. Gromov. Random walk in random groups. Geom. Funct. Anal. 13 (2003), no. 1, 73-146. MR 1978492 (2004j:20088a)
  • [HK] N. Higson, G. Kasparov. $ E$-theory and $ KK$-theory for groups which act properly and isometrically on Hilbert space. Invent. Math. 144 (2001), no. 1, 23-74. MR 1821144 (2002k:19005)
  • [Kr] A. Krieg. Hecke algebras. Mem. Amer. Math. Soc. 87 (1990), no. 435. MR 1027069 (90m:16024)
  • [vdL] H. van der Lek. The homotopy type of complex hyperplane complements, Ph.D. thesis, Univ. of Nijmegen, 1993.
  • [L] Sean Li. Compression bounds for wreath products. Proc. Amer. Math. Soc. 138 (2010), 2701-2714. MR 2644886
  • [N] M. Neuhauser. Relative property (T) and related properties of wreath products. Math. Z. 251 (2005), 167-177. MR 2176470 (2006h:20038)
  • [P] W. Parry, Growth series of some wreath products, Trans. Amer. Math. Soc. 331 (1992), no 2, 751-759. MR 1062874 (92h:20061)
  • [RS] G. Robertson, T. Steger. Negative definite kernels and a dynamical characterization of property T. Ergodic Theory Dynam. Systems 18 (1998), no. 1, 247-253. MR 1609459 (99c:22008)
  • [Ro] J. Roe. Lectures on Coarse Geometry, University Lecture Series, 31. American Mathematical Society, Providence, RI, 2003. MR 2007488 (2004g:53050)
  • [Ru] W. Rudin, Real and complex analysis, McGraw-Hill, 1966. MR 0210528 (35:1420)
  • [T] J. Tits, Buildings of spherical type and finite BN-pairs, Lecture Notes in Math. 386, Springer. MR 0470099 (57:9866)
  • [Z1] E. Zelmanov. Solution of the restricted Burnside problem for groups of odd exponent. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 54, no. 1: 42-59 (1990), translation in Math. USSR-Izv. 36 (1991), no. 1: 41-60. MR 1044047 (91i:20037)
  • [Z2] E. Zelmanov. Solution of the restricted Burnside problem for 2-groups. (Russian) Mat. Sb. 182, no. 4: 568-592 (1991). Translation in Math. USSR-Sb. 72 (1992), no. 2, 543-565. MR 1119009 (93a:20063)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 20F69, 20E22, 43A05, 43A65

Retrieve articles in all journals with MSC (2010): 20F69, 20E22, 43A05, 43A65

Additional Information

Yves Cornulier
Affiliation: Laboratoire de Mathématiques, Bâtiment 425, Université Paris-Sud, 91405 Orsay, France

Yves Stalder
Affiliation: Laboratoire de Mathématiques, UMR 6620-CNRS, Université Blaise Pascal, Campus des Cézeaux, BP 80026, 63171 Aubière Cedex France

Alain Valette
Affiliation: Institut de Mathématiques, Université de Neuchâtel, Rue Émile Argand 11, CP 158, 2009 Neuchâtel, Switzerland

Keywords: Wreath product, measured walls, Haagerup Property, coarse embedding, Kazhdan’s Property T
Received by editor(s): December 8, 2009
Received by editor(s) in revised form: September 14, 2010
Published electronically: February 9, 2012
Additional Notes: The first and second authors were supported by ANR project “QuantiT” (Nr JC08_318197).
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society