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Groups, Geometry, and Dynamics


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Volume 7, Issue 3, 2013, pp. 497–522
DOI: 10.4171/GGD/193

Published online: 2013-08-27

Sharp quantitative nonembeddability of the Heisenberg group into superreflexive Banach spaces

Tim Austin, Assaf Naor[1] and Romain Tessera[2]

(1) New York University, United States
(2) Université Paris-Sud, Orsay, France

Let $\mathbb H$ denote the discrete Heisenberg group, equipped with a word metric $d_W$ associated to some finite symmetric generating set. We show that if $(X,\|\cdot\|)$ is a $p$-convex Banach space then for any Lipschitz function $f\colon \mathbb H\to X$ there exist $x,y\in \mathbb H$ with $d_W(x,y)$ arbitrarily large and \begin{equation} \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) \end{equation} We also show that any embedding into $X$ of a ball of radius $R\ge 4$ in $\mathbb H$ incurs bi-Lipschitz distortion that grows at least as a constant multiple of \begin{equation} \left(\frac{\log R}{\log\log R}\right)^{1/p}. \qquad (2) \end{equation} 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.

Keywords: Bi-Lipschitz embedding, Heisenberg group, superreflexive Banach spaces

Austin Tim, Naor Assaf, Tessera Romain: Sharp quantitative nonembeddability of the Heisenberg group into superreflexive Banach spaces. Groups Geom. Dyn. 7 (2013), 497-522. doi: 10.4171/GGD/193