$L_p$ compression, traveling salesmen, and stable walks

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 $\max \frac{p}{2p-1},\frac{2}{3}$ 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 $\Z\bwr \Z$.

References in corpus (6)

• Euclidean quotients of finite metric spaces
• Differentiating maps into L^1 and the geometry of BV functions
• The wreath product of Z with Z has Hilbert compression exponent 2/3
• Asymptotic isoperimetry on groups and uniform embeddings into Banach spaces
• Compression functions of uniform embeddings of groups into Hilbert and Banach spaces
• The Euclidean distortion of the lamplighter group

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• Distortion in the finite determination result for embeddings of locally finite metric spaces into Banach spaces
• The Haagerup property is not invariant under quasi-isometry
• Group approximation in Cayley topology and coarse geometry, Part I: Coarse embeddings of amenable groups
• On the bi-Lipschitz geometry of lamplighter graphs
• Group approximation in Cayley topology and coarse geometry, Part II: Fibered coarse embeddings
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• Metric $X_p$ inequalities
• Metric Behaviour of the Magnus Embedding
• Law of large numbers for the drift of two-dimensional wreath product
• Compression bounds for wreath products