### 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 1 2009 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Keywords

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

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## Cite this

Austin, T., Naor, A., & Peres, Y. (2009). The wreath product of ℤ with ℤ has Hilbert compression exponent 2/3.

*Proceedings of the American Mathematical Society*,*137*(1), 85-90. https://doi.org/10.1090/S0002-9939-08-09501-4