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 $\Z\bwr \Z$.
Categories: math.MG math.FA math.GR