  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 $\Z\bwr H$
equals $\max{\frac{1}{p},{1/2}}$. We also show that the $L_p$ compression of
$\Z\bwr \Z$ equals $\max{\frac{p}{2p-1},\frac23}$ and the $L_p$ compression of
$(\Z\bwr\Z)_0$ (the zero section of $\Z\bwr \Z$, equipped with the metric
induced from $\Z\bwr \Z$) equals $\max{\frac{p+1}{2p},\frac34}$. The fact that
the Hilbert compression exponent of $\Z\bwr\Z$ equals $\frac23$ while the
Hilbert compression exponent of $(\Z\bwr\Z)_0$ equals $\frac34$ is used to show
that there exists a Lipschitz function $f:(\Z\bwr\Z)_0\to L_2$ which cannot be
extended to a Lipschitz function defined on all of $\Z\bwr \Z$.
