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

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- by Tim Austin, Assaf Naor and Yuval Peres
- Proc. Amer. Math. Soc.
**137**(2009), 85-90 - DOI: https://doi.org/10.1090/S0002-9939-08-09501-4
- Published electronically: August 13, 2008
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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 $\|f(x)-f(y)\|_2\ge cd(x,y)^\alpha .$ It was previously known that the Hilbert compression exponent of the wreath product $\mathbb {Z} {\boldsymbol {\wr }} \mathbb {Z}$ is between $\frac 23$ and $\frac 34$. Here we show that $\frac 23$ is the correct value. Our proof is based on an application of K. Ball’s notion of Markov type.## References

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## Bibliographic 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
- MR Author ID: 137920
- 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
- © Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**137**(2009), 85-90 - MSC (2000): Primary 20F65, 51F99
- DOI: https://doi.org/10.1090/S0002-9939-08-09501-4
- MathSciNet review: 2439428