JournalsggdVol. 7, No. 3pp. 497–522

Sharp quantitative nonembeddability of the Heisenberg group into superreflexive Banach spaces

  • Tim Austin

    New York University, USA
  • Assaf Naor

    New York University, United States
  • Romain Tessera

    Université Paris-Sud, Orsay, France
Sharp quantitative nonembeddability of the Heisenberg group into superreflexive Banach spaces cover
Download PDF

Abstract

Let H\mathbb H denote the discrete Heisenberg group, equipped with a word metric dWd_W associated to some finite symmetric generating set. We show that if (X,)(X,\|\cdot\|) is a pp-convex Banach space then for any Lipschitz function f ⁣:HXf\colon \mathbb H\to X there exist x,yHx,y\in \mathbb H with dW(x,y)d_W(x,y) arbitrarily large and

f(x)f(y)dW(x,y)(loglogdW(x,y)logdW(x,y))1/p.(1)\frac{\|f(x)-f(y)\|}{d_W(x,y)}\lesssim \bigg(\frac{\log\log d_W(x,y)}{\log d_W(x,y)}\bigg)^{1/p}. \qquad (1)

We also show that any embedding into XX of a ball of radius R4R\ge 4 in H\mathbb H incurs bi-Lipschitz distortion that grows at least as a constant multiple of

(logRloglogR)1/p.(2)\left(\frac{\log R}{\log\log R}\right)^{1/p}. \qquad (2)

Both (1) and (2) are sharp up to the iterated logarithm terms. When XX is Hilbert space we obtain a representation-theoretic proof yielding bounds corresponding to (1) and (2) which are sharp up to a universal constant.

Cite this article

Tim Austin, Assaf Naor, Romain Tessera, Sharp quantitative nonembeddability of the Heisenberg group into superreflexive Banach spaces. Groups Geom. Dyn. 7 (2013), no. 3, pp. 497–522

DOI 10.4171/GGD/193