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\\).<\/p>\n","protected":false},"excerpt":{"rendered":"
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 the compression of (the zero section of , equipped with the metric induced from ) equals . 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