$L_p$ compression, traveling salesmen, and stable walks

TL;DR

This paper investigates the $L_p$ compression exponents of certain wreath products involving groups of polynomial growth, revealing precise values and implications for Lipschitz extension problems.

Contribution

It provides exact calculations of $L_p$ compression exponents for wreath products of groups with polynomial growth, and demonstrates a Lipschitz extension obstruction.

Findings

$L_p$ compression of $ ext{Z} wr H$ equals $ ext{max}(rac{1}{p}, rac{1}{2})$ for polynomial growth groups $H$ with growth rate ≥ 2.

$L_p$ compression of $ ext{Z} wr ext{Z}$ equals $ ext{max}(rac{p}{2p-1}, rac{2}{3})$.

There exists a Lipschitz function on $( ext{Z} wr ext{Z})_0$ that cannot be extended to $ ext{Z} wr ext{Z}$.

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 $\Z\bwr H$ equals $\max{\frac{1}{p},{1/2}}$. We also show that the $L_p$ compression of $\Z\bwr \Z$ equals $\max{\frac{p}{2p-1},\frac23}$ 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 $\max{\frac{p+1}{2p},\frac34}$. The fact that the Hilbert compression exponent of $\Z\bwr\Z$ equals $\frac23$ while the Hilbert compression exponent of $(\Z\bwr\Z)_0$ equals $\frac34$ 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 $\Z\bwr \Z$.