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

TL;DR

This paper determines the exact Hilbert compression exponent of the wreath product of Z with Z as 2/3, resolving a previously known upper and lower bound gap using Markov type techniques.

Contribution

It proves that the Hilbert compression exponent of Z wreath Z is exactly 2/3, confirming the conjectured value and refining prior bounds.

Findings

Hilbert compression exponent of Z wreath Z is 2/3

Uses Markov type to establish the exact exponent

Resolves a gap between previous bounds

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 $\|f(x)-f(y)\|_2\ge cd(x,y)^\alpha.$ 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.