### Abstract

Let H denote the discrete Heisenberg group, equipped with a word metric dW associated to some finite symmetric generating set. We show that if (X, ∥ · ∥) is a p-convex Banach space then for any Lipschitz function f : ℍ → X there exist x; ⋯ ℍ with d_{W} (x, y) arbitrarily large and (eqution presented) We also show that any embedding into X of a ball of radius R ≥ 4 in ℍ incurs bi-Lipschitz distortion that grows at least as a constant multiple of (eqution presented) Both (1) and (2) are sharp up to the iterated logarithm terms. When X is Hilbert space we obtain a representation-theoretic proof yielding bounds corresponding to (1) and (2) which are sharp up to a universal constant.

Original language | English (US) |
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Pages (from-to) | 497-522 |

Number of pages | 26 |

Journal | Groups, Geometry, and Dynamics |

Volume | 7 |

Issue number | 3 |

DOIs | |

State | Published - Oct 25 2013 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Geometry and Topology
- Discrete Mathematics and Combinatorics

### Keywords

- Bi-Lipschitz embedding
- Heisenberg group
- Superreflexive Banach spaces

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## Cite this

*Groups, Geometry, and Dynamics*,

*7*(3), 497-522. https://doi.org/10.4171/GGD/193