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 Lp compression of the wreath product Z oH equals max { 1 p , 1 2 } . We also show that the Lp compression of Z oZ equals max { p 2p−1 , 2 3 } and the Lp compression of (Z oZ)0 (the zero section of Z oZ, equipped with the metric induced from Z o Z) equals max { p+1 2p , 3 4 } . The fact that the Hilbert compression exponent of Z o Z equals 3 while the Hilbert compression exponent of (Z o Z)0 equals 3 4 is used to show that there exists a Lipschitz function f : (Z o Z)0 → L2 which cannot be extended to a Lipschitz function defined on all of Z o Z.