The wreath product of Z with Z has Hilbert compression exponent 2/3

Austin, Tim
Naor, Assaf
Peres, Yuval
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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 $\|f(x)-f(y)\|_2\ge cd(x,y)^\alpha.$ In \cite{AGS06} it was shown that the Hilbert compression exponent of the wreath product $\Z\bwr \Z $ is at most $\frac34$, and in \cite{NP07} was proved that this exponent is at least $\frac23$. Here we show that $\frac23$ is the correct value. Our proof is based on an application of K. Ball's notion of Markov type.
Comment: Removed a reference to the lower bound of 2/3 for the Hilbert compression of Z wreath Z in math/0603138 since the proof is incorrect; added a reference which contains a correct proof (the results of this paper remain unchanged). Added Remark 2.2 which shows why Z wreath Z has Hilbert compression exponent at least 2/3
Mathematics - Metric Geometry, Mathematics - Functional Analysis, Mathematics - Group Theory