Lp compression, traveling salesmen, and stable walks

Abstract

We show that if $H$ is a group of polynomial growth whose growth rate is at least quadratic, then the $L_p$ compression of the wreath product $\mathbb{Z}\wr H$ equals ${\displaystyle \max \{\frac{1}{p},\frac{1}{2}\}}$. We also show that the $L_p$ compression of $\mathbb{Z}\wr \mathbb{Z}$ equals ${\displaystyle \max \{\frac{p}{2p-1},\frac{2}{3}\}}$ and that the $L_p$ compression of $(\mathbb{Z}\wr \mathbb{Z}{)}_0$ (the zero section of $\mathbb{Z}\wr \mathbb{Z}$, equipped with the metric induced from $\mathbb{Z}\wr \mathbb{Z}$) equals ${\displaystyle \max \{\frac{p+1}{2p},\frac{3}{4}\}}$. The fact that the Hilbert compression exponent of $\mathbb{Z}\wr \mathbb{Z}$ equals $2/3$ while the Hilbert compression exponent of $(\mathbb{Z}\wr \mathbb{Z}{)}_0$ equals $3/4$ is used to show that there exists a Lipschitz function $f:(\mathbb{Z}\wr \mathbb{Z}{)}_0\to L_2$ which cannot be extended to a Lipschitz function defined on all of $\mathbb{Z}\wr \mathbb{Z}$.