### 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{double-struck} {wreath product} H equals max. We also show that the L
_{p} compression of Z{double-struck} {wreath product} Z{double-struck} equals max and that the L
_{p} compression of(Z{double-struck} {wreath product} Z{double-struck})
_{0} (the zero section of Z{double-struck} {wreath product} Z{double-struck}, equipped with the metric induced from Z{double-struck} {wreath product} Z) equals max. The fact that the Hilbert compression exponent of Z{double-struck} {wreath product} Z{double-struck} equals 2/3 while the Hilbert compression exponent of (Z{double-struck} {wreath product} Z{double-struck})
_{0} equals 3/4 is used to show that there exists a Lipschitz function f : (Z{double-struck} {wreath product} Z{double-struck})
_{0} → L
_{2} which cannot be extended to a Lipschitz function defined on all of Z{double-struck} {wreath product} Z{double-struck}.

Original language | English (US) |
---|---|

Pages (from-to) | 53-108 |

Number of pages | 56 |

Journal | Duke Mathematical Journal |

Volume | 157 |

Issue number | 1 |

DOIs | |

State | Published - Mar 15 2011 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Duke Mathematical Journal*,

*157*(1), 53-108. https://doi.org/10.1215/00127094-2011-002

**L
p compression, traveling salesmen, and stable walks.** / Naor, Assaf; Peres, Yuval.

Research output: Contribution to journal › Article

*Duke Mathematical Journal*, vol. 157, no. 1, pp. 53-108. https://doi.org/10.1215/00127094-2011-002

}

TY - JOUR

T1 - L p compression, traveling salesmen, and stable walks

AU - Naor, Assaf

AU - Peres, Yuval

PY - 2011/3/15

Y1 - 2011/3/15

N2 - 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{double-struck} {wreath product} H equals max. We also show that the L p compression of Z{double-struck} {wreath product} Z{double-struck} equals max and that the L p compression of(Z{double-struck} {wreath product} Z{double-struck}) 0 (the zero section of Z{double-struck} {wreath product} Z{double-struck}, equipped with the metric induced from Z{double-struck} {wreath product} Z) equals max. The fact that the Hilbert compression exponent of Z{double-struck} {wreath product} Z{double-struck} equals 2/3 while the Hilbert compression exponent of (Z{double-struck} {wreath product} Z{double-struck}) 0 equals 3/4 is used to show that there exists a Lipschitz function f : (Z{double-struck} {wreath product} Z{double-struck}) 0 → L 2 which cannot be extended to a Lipschitz function defined on all of Z{double-struck} {wreath product} Z{double-struck}.

AB - 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{double-struck} {wreath product} H equals max. We also show that the L p compression of Z{double-struck} {wreath product} Z{double-struck} equals max and that the L p compression of(Z{double-struck} {wreath product} Z{double-struck}) 0 (the zero section of Z{double-struck} {wreath product} Z{double-struck}, equipped with the metric induced from Z{double-struck} {wreath product} Z) equals max. The fact that the Hilbert compression exponent of Z{double-struck} {wreath product} Z{double-struck} equals 2/3 while the Hilbert compression exponent of (Z{double-struck} {wreath product} Z{double-struck}) 0 equals 3/4 is used to show that there exists a Lipschitz function f : (Z{double-struck} {wreath product} Z{double-struck}) 0 → L 2 which cannot be extended to a Lipschitz function defined on all of Z{double-struck} {wreath product} Z{double-struck}.

UR - http://www.scopus.com/inward/record.url?scp=79953230922&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79953230922&partnerID=8YFLogxK

U2 - 10.1215/00127094-2011-002

DO - 10.1215/00127094-2011-002

M3 - Article

VL - 157

SP - 53

EP - 108

JO - Duke Mathematical Journal

JF - Duke Mathematical Journal

SN - 0012-7094

IS - 1

ER -