Sharp quantitative nonembeddability of the Heisenberg group into superreflexive Banach spaces

Tim Austin, Assaf Naor, Romain Tessera

Research output: Contribution to journalArticle

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 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.

Original languageEnglish (US)
Pages (from-to)497-522
Number of pages26
JournalGroups, Geometry, and Dynamics
Volume7
Issue number3
DOIs
StatePublished - 2013

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Heisenberg Group
Banach space
Generating Set
Lipschitz Function
Discrete Group
Logarithm
Lipschitz
Ball
Hilbert space
Radius
Denote
Metric
Term

Keywords

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

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Geometry and Topology

Cite this

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

In: Groups, Geometry, and Dynamics, Vol. 7, No. 3, 2013, p. 497-522.

Research output: Contribution to journalArticle

Austin, Tim ; Naor, Assaf ; Tessera, Romain. / Sharp quantitative nonembeddability of the Heisenberg group into superreflexive Banach spaces. In: Groups, Geometry, and Dynamics. 2013 ; Vol. 7, No. 3. pp. 497-522.
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