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Proceedings of the American Mathematical Society
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
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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.


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Additional Information

Tim Austin
Affiliation: Department of Mathematics, University of California at Los Angeles, Los Angeles, California 90095
Email: timaustin@math.ucla.edu

Assaf Naor
Affiliation: Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, New York 10012
Email: naor@cims.nyu.edu

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

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09501-4
PII: S 0002-9939(08)09501-4
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.