# 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

## Abstract

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

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

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.

## 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