Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



The wreath product of $ \mathbb{Z}$ with $ \mathbb{Z}$ has Hilbert compression exponent $ \frac{2}{3}$

Authors: Tim Austin, Assaf Naor and Yuval Peres
Journal: Proc. Amer. Math. Soc. 137 (2009), 85-90
MSC (2000): Primary 20F65, 51F99
Published electronically: August 13, 2008
MathSciNet review: 2439428
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ G$ be a finitely generated group, equipped with the word metric $ d$ associated with some finite set of generators. The Hilbert compression exponent of $ G$ is the supremum over all $ \alpha\ge 0$ such that there exists a Lipschitz mapping $ f:G\to L_2$ and a constant $ c>0$ such that for all $ x,y\in G$ we have $ \Vert f(x)-f(y)\Vert _2\ge cd(x,y)^\alpha.$ It was previously known that the Hilbert compression exponent of the wreath product $ \mathbb{Z}\,{\boldsymbol{\wr}}\, \mathbb{Z} $ is between $ \frac23$ and $ \frac34$. Here we show that $ \frac23$ is the correct value. Our proof is based on an application of K. Ball's notion of Markov type.

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

  • 1. G. Arzhantseva, C. Drutu, and M. Sapir.
    Compression functions of uniform embeddings of groups into Hilbert and Banach spaces.
    Preprint, 2006. Available at
  • 2. 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(4):911-929, 2006. MR 2271228 (2007k:20084)
  • 3. K. Ball.
    Markov chains, Riesz transforms and Lipschitz maps.
    Geom. Funct. Anal., 2(2):137-172, 1992. MR 1159828 (93b:46025)
  • 4. Y. Bartal, N. Linial, M. Mendel, and A. Naor.
    On metric Ramsey-type phenomena.
    Ann. of Math. (2), 162(2):643-709, 2005. MR 2183280 (2006g:46035)
  • 5. Y. de Cornulier, R. Tessera, and A. Valette.
    Isometric group actions on Hilbert spaces: growth of cocycles. Geom. Funct. Anal., 17(3):770-792, 2007. MR 2346274
  • 6. A. G. Èrshler.
    On the asymptotics of the rate of departure to infinity.
    Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 283 (Teor. Predst. Din. Sist. Komb. i Algoritm. Metody. (6):251-257, 263, 2001. MR 1879073 (2003a:60065)
  • 7. E. Guentner and J. Kaminker.
    Exactness and uniform embeddability of discrete groups.
    J. London Math. Soc. (2), 70(3):703-718, 2004. MR 2160829 (2006i:43006)
  • 8. V. A. Kaĭmanovich and A. M. Vershik.
    Random walks on discrete groups: boundary and entropy.
    Ann. Probab., 11(3):457-490, 1983. MR 704539 (85d:60024)
  • 9. N. Linial, A. Magen, and A. Naor.
    Girth and Euclidean distortion.
    Geom. Funct. Anal., 12(2):380-394, 2002. MR 1911665 (2003d:05054)
  • 10. M. Mendel and A. Naor.
    Some applications of Ball's extension theorem.
    Proc. Amer. Math. Soc., 134(9):2577-2584 (electronic), 2006. MR 2213735 (2007a:46014)
  • 11. A. Naor.
    A phase transition phenomenon between the isometric and isomorphic extension problems for Hölder functions between $ L\sb p$ spaces.
    Mathematika, 48(1-2):253-271 (2003), 2001. MR 1996375 (2004f:46013)
  • 12. A. Naor and Y. Peres.
    Embeddings of discrete groups and the speed of random walks.
    To appear in Internat. Math. Res. Notices. Available at
  • 13. A. Naor, Y. Peres, O. Schramm, and S. Sheffield.
    Markov chains in smooth Banach spaces and Gromov-hyperbolic metric spaces.
    Duke Math. J., 134(1):165-197, 2006. MR 2239346 (2007k:46017)
  • 14. A. L. T. Paterson.
    Amenability, volume 29 of Mathematical Surveys and Monographs.
    American Mathematical Society, Providence, RI, 1988. MR 961261 (90e:43001)
  • 15. D. Revelle.
    Rate of escape of random walks on wreath products and related groups.
    Ann. Probab., 31(4):1917-1934, 2003. MR 2016605 (2005a:60070)
  • 16. R. Tessera.
    Asymptotic isoperimetry on groups and uniform embeddings into Banach spaces.
    Preprint, 2006. Available at
  • 17. J. H. Wells and L. R. Williams.
    Embeddings and extensions in analysis.
    Springer-Verlag, New York, 1975.
    Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 84. MR 0461107 (57:1092)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 20F65, 51F99

Retrieve articles in all journals with MSC (2000): 20F65, 51F99

Additional Information

Tim Austin
Affiliation: Department of Mathematics, University of California at Los Angeles, Los Angeles, California 90095

Assaf Naor
Affiliation: Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, New York 10012

Yuval Peres
Affiliation: Microsoft Research, One Microsoft Way, Redmond, Washington 98052 – and – Department of Mathematics, University of California, Berkeley, California 94720-3840

Keywords: Geometric group theory, coarse geometry, Markov type, Hilbert compression exponents
Received by editor(s): June 13, 2007
Received by editor(s) in revised form: January 3, 2008
Published electronically: August 13, 2008
Additional Notes: The second author was supported in part by NSF grants CCF-0635078 and DMS-0528387.
The third author was supported in part by NSF grant DMS-0605166.
Communicated by: Alexander N. Dranishnikov
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society