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

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 $\mathrm{\parallel }f(x)-f(y){\mathrm{\parallel }}_2\ge cd(x,y{)}^{\alpha }\mathrm{.}$ In \cite{AGS06} it was shown that the Hilbert compression exponent of the wreath product $\mathbb{Z}\text{bwr}\mathbb{Z}$ is at most $\frac{3}{4}$, and in \cite{NP07} was proved that this exponent is at least $\frac{2}{3}$. Here we show that $\frac{2}{3}$ is the correct value. Our proof is based on an application of K. Ball’s notion of Markov type.