## Abstract

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 $\mathrm{max}\{\frac{1}{p},\frac{1}{2}\}$. We also show that the ${L}_{p}$ compression of $\mathbb{Z}\wr \mathbb{Z}$ equals $\mathrm{max}\{\frac{p}{2p-1},\frac{2}{3}\}$ 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 $\mathrm{max}\{\frac{p+1}{2p},\frac{3}{4}\}$. 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 $f:(\mathbb{Z}\wr \mathbb{Z}{)}_{0}\to {L}_{2}$ which cannot be extended to a Lipschitz function defined on all of $\mathbb{Z}\wr \mathbb{Z}$.

## Citation

Assaf Naor. Yuval Peres. "${L}_{p}$ compression, traveling salesmen, and stable walks." Duke Math. J. 157 (1) 53 - 108, 15 March 2011. https://doi.org/10.1215/00127094-2011-002

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