### 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) |
---|---|

Pages (from-to) | 497-522 |

Number of pages | 26 |

Journal | Groups, Geometry, and Dynamics |

Volume | 7 |

Issue number | 3 |

DOIs | |

State | Published - 2013 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Geometry and Topology

### Cite this

*Groups, Geometry, and Dynamics*,

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

**Sharp quantitative nonembeddability of the Heisenberg group into superreflexive Banach spaces.** / Austin, Tim; Naor, Assaf; Tessera, Romain.

Research output: Contribution to journal › Article

*Groups, Geometry, and Dynamics*, vol. 7, no. 3, pp. 497-522. https://doi.org/10.4171/GGD/193

}

TY - JOUR

T1 - Sharp quantitative nonembeddability of the Heisenberg group into superreflexive Banach spaces

AU - Austin, Tim

AU - Naor, Assaf

AU - Tessera, Romain

PY - 2013

Y1 - 2013

N2 - 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 dW (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.

AB - 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 dW (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.

KW - Bi-Lipschitz embedding

KW - Heisenberg group

KW - Superreflexive Banach spaces

UR - http://www.scopus.com/inward/record.url?scp=84886899092&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84886899092&partnerID=8YFLogxK

U2 - 10.4171/GGD/193

DO - 10.4171/GGD/193

M3 - Article

VL - 7

SP - 497

EP - 522

JO - Groups, Geometry, and Dynamics

JF - Groups, Geometry, and Dynamics

SN - 1661-7207

IS - 3

ER -