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 $(X,\Vert \cdot \Vert )$ is a $p$-convex Banach space then for any Lipschitz function $f:\mathbb{H}\to X$ 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.