Compression bounds for wreath products

Author:
Sean Li

Journal:
Proc. Amer. Math. Soc. **138** (2010), 2701-2714

MSC (2010):
Primary 20F65, 51F99

Published electronically:
April 5, 2010

MathSciNet review:
2644886

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Abstract: We show that if and are finitely generated groups whose Hilbert compression exponent is positive, then so is the Hilbert compression exponent of the wreath product . We also prove an analogous result for coarse embeddings of wreath products. In the special case , our result implies that the Hilbert compression exponent of is at least , answering a question posed by several authors.

**[AGS06]**G. N. Arzhantseva, V. S. Guba, and M. V. Sapir,*Metrics on diagram groups and uniform embeddings in a Hilbert space*, Comment. Math. Helv.**81**(2006), no. 4, 911–929. MR**2271228**, 10.4171/CMH/80**[ANP09]**Tim Austin, Assaf Naor, and Yuval Peres,*The wreath product of ℤ with ℤ has Hilbert compression exponent \frac{2}3*, Proc. Amer. Math. Soc.**137**(2009), no. 1, 85–90. MR**2439428**, 10.1090/S0002-9939-08-09501-4**[BL00]**Yoav Benyamini and Joram Lindenstrauss,*Geometric nonlinear functional analysis. Vol. 1*, American Mathematical Society Colloquium Publications, vol. 48, American Mathematical Society, Providence, RI, 2000. MR**1727673****[CCJ**Pierre-Alain Cherix, Michael Cowling, Paul Jolissaint, Pierre Julg, and Alain Valette,^{+}01]*Groups with the Haagerup property*, Progress in Mathematics, vol. 197, Birkhäuser Verlag, Basel, 2001. Gromov’s a-T-menability. MR**1852148****[CK06]**J. Cheeger and B. Kleiner,*Differentiating maps into and the geometry of BV functions*, to appear in Ann. of Math.**[dCSV09]**Y. de Cornulier, Y. Stalder, and A. Valette,*Proper actions of wreath products and generalizations*, preprint, 2009.**[DL97]**Michel Marie Deza and Monique Laurent,*Geometry of cuts and metrics*, Algorithms and Combinatorics, vol. 15, Springer-Verlag, Berlin, 1997. MR**1460488****[FJ03]**Richard J. Fleming and James E. Jamison,*Isometries on Banach spaces: function spaces*, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 129, Chapman & Hall/CRC, Boca Raton, FL, 2003. MR**1957004****[Gal08]**Światosław R. Gal,*Asymptotic dimension and uniform embeddings*, Groups Geom. Dyn.**2**(2008), no. 1, 63–84. MR**2367208**, 10.4171/GGD/31**[GK04]**Erik Guentner and Jerome Kaminker,*Exactness and uniform embeddability of discrete groups*, J. London Math. Soc. (2)**70**(2004), no. 3, 703–718. MR**2160829**, 10.1112/S0024610704005897**[Lam58]**John Lamperti,*On the isometries of certain function-spaces*, Pacific J. Math.**8**(1958), 459–466. MR**0105017****[Mat02]**Jiří Matoušek,*Lectures on discrete geometry*, Graduate Texts in Mathematics, vol. 212, Springer-Verlag, New York, 2002. MR**1899299****[NP08]**Assaf Naor and Yuval Peres,*Embeddings of discrete groups and the speed of random walks*, Int. Math. Res. Not. IMRN , posted on (2008), Art. ID rnn 076, 34. MR**2439557**, 10.1093/imrn/rnn076**[NP09]**-,*compression, traveling salesmen, and stable walks*, preprint.**[SV07]**Yves Stalder and Alain Valette,*Wreath products with the integers, proper actions and Hilbert space compression*, Geom. Dedicata**124**(2007), 199–211. MR**2318545**, 10.1007/s10711-006-9119-3**[Tes09]**R. Tessera,*Asymptotic isoperimetry on groups and uniform embeddings into Banach spaces*, to appear in Comment. Math. Helv., 2009.**[Woj91]**P. Wojtaszczyk,*Banach spaces for analysts*, Cambridge Studies in Advanced Mathematics, vol. 25, Cambridge University Press, Cambridge, 1991. MR**1144277****[WW75]**J. H. Wells and L. R. Williams,*Embeddings and extensions in analysis*, Springer-Verlag, New York-Heidelberg, 1975. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 84. MR**0461107**

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Additional Information

**Sean Li**

Affiliation:
Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, New York 10012-1185

Email:
seanli@cims.nyu.edu

DOI:
https://doi.org/10.1090/S0002-9939-10-10307-4

Received by editor(s):
September 2, 2009

Received by editor(s) in revised form:
December 3, 2009

Published electronically:
April 5, 2010

Additional Notes:
This work was supported in part by NSF grants CCF-0635078 and CCF-0832795.

Communicated by:
Alexander N. Dranishnikov

Article copyright:
© Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.