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 L_p compression of the wreath product Z{double-struck} {wreath product} H equals max. We also show that the L_p compression of Z{double-struck} {wreath product} Z{double-struck} equals max and that the L_p compression of(Z{double-struck} {wreath product} Z{double-struck})₀ (the zero section of Z{double-struck} {wreath product} Z{double-struck}, equipped with the metric induced from Z{double-struck} {wreath product} Z) equals max. The fact that the Hilbert compression exponent of Z{double-struck} {wreath product} Z{double-struck} equals 2/3 while the Hilbert compression exponent of (Z{double-struck} {wreath product} Z{double-struck})₀ equals 3/4 is used to show that there exists a Lipschitz function f : (Z{double-struck} {wreath product} Z{double-struck})₀ → L₂ which cannot be extended to a Lipschitz function defined on all of Z{double-struck} {wreath product} Z{double-struck}.