## Abstract

Let ℍ = 〈a, b|a[a, b] = [a, b]a ∧ b[a, b] = [a, b]b〉 be the discrete Heisenberg group, equipped with the left-invariant word metric d_{W}(·, ·) associated to the generating set {a, b, a^{−1}, b^{−1}}. Letting B_{n} = {x ∈ ℍ: d_{W}(x, e_{ℍ}) ⩽ n} denote the corresponding closed ball of radius n ∈ ℕ, and writing c = [a, b] = aba^{−1}b^{−1}, we prove that if (X, ‖ · ‖X) is a Banach space whose modulus of uniform convexity has power type q ∈ [2,∞), then there exists K ∈ (0, ∞) such that every f: ℍ → X satisfies (Formula Presented). It follows that for every n ∈ ℕ the bi-Lipschitz distortion of every f: B_{n} → X is at least a constant multiple of (log n)^{1/q}, an asymptotically optimal estimate as n → ∞.

Original language | English (US) |
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Pages (from-to) | 309-339 |

Number of pages | 31 |

Journal | Israel Journal of Mathematics |

Volume | 203 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2014 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Mathematics(all)