Sharp quantitative nonembeddability of the Heisenberg group into superreflexive Banach spaces

Tim Austin, Assaf Naor, Romain Tessera

Research output: Contribution to journalArticle

11 Scopus citations

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 - Oct 25 2013
Externally publishedYes

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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