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

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.