LP 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 \(\Z\bwr H\) equals \(max\frac{1}{p},1/2\). We also show that the \(L_p\)compression of \(\Z\bwr \Z\) equals \(max\frac{p}{2p-1},\frac{2}{3}\) 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 \(max\frac{p+1}{2p},\frac{3}{4}\). The fact that the Hilbert compression exponent of \(\Z\bwr\Z\) equals \(\frac{2}{3}\) while the Hilbert compression exponent of \((\Z\bwr\Z)_0\) equals \(\frac{3}{4}\) 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 \(\Z\bwr \Z\).