We show that if is a group of polynomial growth whose growth rate is at least quadratic, then the compression of the wreath product equals . We also show that the compression of equals and that the compression of (the zero section of , equipped with the metric induced from ) equals . The fact that the Hilbert compression exponent of equals while the Hilbert compression exponent of equals is used to show that there exists a Lipschitz function which cannot be extended to a Lipschitz function defined on all of .
Assaf Naor. Yuval Peres. " compression, traveling salesmen, and stable walks." Duke Math. J. 157 (1) 53 - 108, 15 March 2011. https://doi.org/10.1215/00127094-2011-002