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 Lp compression of the wreath product $\Z\bwr H$ equals max1p,1/2. We also show that the Lpcompression of $\Z\bwr \Z$ equals maxp2p−1,23 and the Lp compression of $(\Z\bwr\Z)_0$ (the zero section of $\Z\bwr \Z$, equipped with the metric induced from $\Z\bwr \Z$) equals maxp+12p,34. The fact that the Hilbert compression exponent of $\Z\bwr\Z$ equals 23 while the Hilbert compression exponent of $(\Z\bwr\Z)_0$ equals 34 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 $\Z\bwr \Z$.