Sharp quantitative nonembeddability of the Heisenberg group into superreflexive Banach spaces

TL;DR

This paper establishes sharp quantitative bounds on how poorly the discrete Heisenberg group can be embedded into superreflexive Banach spaces, revealing fundamental geometric obstructions.

Contribution

It provides the first sharp quantitative nonembeddability results for the Heisenberg group into superreflexive Banach spaces, including new bounds for Lipschitz and bi-Lipschitz embeddings.

Findings

Lipschitz embeddings distort distances by at least a logarithmic factor.

Embedding a ball of radius R incurs distortion growing as (log R / log log R)^{1/p}.

Results are sharp up to iterated logarithm terms.

Abstract

Let $\H$ 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 $f:\H\to X$ there exist $x,y\in \H$ with $d_W(x,y)$ arbitrarily large and \begin{equation}\label{eq:comp abs} \frac{\|f(x)-f(y)\|}{d_W(x,y)}\lesssim \left(\frac{\log\log d_W(x,y)}{\log d_W(x,y)}\right)^{1/p}. \end{equation} We also show that any embedding into $X$ of a ball of radius $R\ge 4$ in $\H$ incurs bi-Lipschitz distortion that grows at least as a constant multiple of \begin{equation}\label{eq:dist abs} \left(\frac{\log R}{\log\log R}\right)^{1/p}. \end{equation} Both~\eqref{eq:comp abs} and~\eqref{eq:dist abs} are sharp up to the iterated logarithm terms. When $X$ is Hilbert space we obtain a representation-theoretic proof yielding bounds corresponding to~\eqref{eq:comp abs} and~\eqref{eq:dist abs} which are sharp up to a universal constant.