Naor, Assaf; Peres, Yuval \(L_p\) compression, traveling salesmen, and stable walks. (English) Zbl 1268.20044 Duke Math. J. 157, No. 1, 53-108 (2011). Summary: 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 \(\mathbb Z\wr H\) equals \(\max\{\tfrac 1p,\tfrac 12\}\). We also show that the \(L_p\) compression of \(\mathbb Z\wr\mathbb Z\) equals \(\max\{\tfrac p{2p-1},\tfrac 23\}\) and that the \(L_p\) compression of \((\mathbb Z\wr\mathbb Z)_0\) (the zero section of \(\mathbb Z\wr\mathbb Z\), equipped with the metric induced from \(\mathbb Z\wr\mathbb Z\)) equals \(\max\{\tfrac{p+1}{2p},\tfrac 34\}\). The fact that the Hilbert compression exponent of \(\mathbb Z\wr\mathbb Z\) equals \(2/3\) while the Hilbert compression exponent of \((\mathbb Z\wr\mathbb Z)_0\) equals \(3/4\) is used to show that there exists a Lipschitz function \(f\colon(\mathbb Z\wr\mathbb Z)_0\to L_2\) which cannot be extended to a Lipschitz function defined on all of \(\mathbb Z\wr\mathbb Z\). Cited in 1 ReviewCited in 30 Documents MSC: 20F65 Geometric group theory 46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product) 43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc. 43A07 Means on groups, semigroups, etc.; amenable groups 60G50 Sums of independent random variables; random walks Keywords:groups of polynomial growth; growth rates; \(L_p\)-compression; wreath products; Hilbert compression exponents; Lipschitz functions × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] I. Aharoni, B. Maurey, and B. S. 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