Sharp quantitative nonembeddability of the Heisenberg group into superreflexive Banach spaces

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.