$L_p$ compression, traveling salesmen, and stable walks

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 $\mathbb{Z}\text{bwr}H$ equals $\max \frac{1}{p},1\mathrm{/}2$. We also show that the $L_p$ compression of $\mathbb{Z}\text{bwr}\mathbb{Z}$ equals $\max \frac{p}{2p-1},\frac{2}{3}$ and the $L_p$ compression of $(\mathbb{Z}\text{bwr}\mathbb{Z}{)}_0$ (the zero section of $\mathbb{Z}\text{bwr}\mathbb{Z}$, equipped with the metric induced from $\mathbb{Z}\text{bwr}\mathbb{Z}$) equals $\max \frac{p+1}{2p},\frac{3}{4}$. The fact that the Hilbert compression exponent of $\mathbb{Z}\text{bwr}\mathbb{Z}$ equals $\frac{2}{3}$ while the Hilbert compression exponent of $(\mathbb{Z}\text{bwr}\mathbb{Z}{)}_0$ equals $\frac{3}{4}$ is used to show that there exists a Lipschitz function $f:(\mathbb{Z}\text{bwr}\mathbb{Z}{)}_0\to L_2$ which cannot be extended to a Lipschitz function defined on all of $\mathbb{Z}\text{bwr}\mathbb{Z}$.