### Abstract

Let G be a finitely generated group, equipped with the word metric d associated with some finite set of generators. The Hilbert compression exponent of G is the supremum over all α ≥ 0 such that there exists a Lipschitz mapping f : G → L _{2} and a constant c > 0 such that for all x,y ∈ G we have ||f(x) - f(y)|| _{2} ≥ cd(x,y) ^{α}. It was previously known that the Hilbert compression exponent of the wreath product ℤ ∼ ℤ is between 2/3 and 3/4. Here we show that 2/3 is the correct value. Our proof is based on an application of K. Ball's notion of Markov type.

Original language | English (US) |
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Pages (from-to) | 85-90 |

Number of pages | 6 |

Journal | Proceedings of the American Mathematical Society |

Volume | 137 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2009 |

### Fingerprint

### Keywords

- Coarse geometry
- Geometric group theory
- Hilbert compression exponents
- Markov type

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Proceedings of the American Mathematical Society*,

*137*(1), 85-90. https://doi.org/10.1090/S0002-9939-08-09501-4

**The wreath product of ℤ with ℤ has Hilbert compression exponent 2/3.** / Austin, Tim; Naor, Assaf; Peres, Yuval.

Research output: Contribution to journal › Article

*Proceedings of the American Mathematical Society*, vol. 137, no. 1, pp. 85-90. https://doi.org/10.1090/S0002-9939-08-09501-4

}

TY - JOUR

T1 - The wreath product of ℤ with ℤ has Hilbert compression exponent 2/3

AU - Austin, Tim

AU - Naor, Assaf

AU - Peres, Yuval

PY - 2009/1

Y1 - 2009/1

N2 - Let G be a finitely generated group, equipped with the word metric d associated with some finite set of generators. The Hilbert compression exponent of G is the supremum over all α ≥ 0 such that there exists a Lipschitz mapping f : G → L 2 and a constant c > 0 such that for all x,y ∈ G we have ||f(x) - f(y)|| 2 ≥ cd(x,y) α. It was previously known that the Hilbert compression exponent of the wreath product ℤ ∼ ℤ is between 2/3 and 3/4. Here we show that 2/3 is the correct value. Our proof is based on an application of K. Ball's notion of Markov type.

AB - Let G be a finitely generated group, equipped with the word metric d associated with some finite set of generators. The Hilbert compression exponent of G is the supremum over all α ≥ 0 such that there exists a Lipschitz mapping f : G → L 2 and a constant c > 0 such that for all x,y ∈ G we have ||f(x) - f(y)|| 2 ≥ cd(x,y) α. It was previously known that the Hilbert compression exponent of the wreath product ℤ ∼ ℤ is between 2/3 and 3/4. Here we show that 2/3 is the correct value. Our proof is based on an application of K. Ball's notion of Markov type.

KW - Coarse geometry

KW - Geometric group theory

KW - Hilbert compression exponents

KW - Markov type

UR - http://www.scopus.com/inward/record.url?scp=69049083270&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=69049083270&partnerID=8YFLogxK

U2 - 10.1090/S0002-9939-08-09501-4

DO - 10.1090/S0002-9939-08-09501-4

M3 - Article

AN - SCOPUS:69049083270

VL - 137

SP - 85

EP - 90

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 1

ER -