Sharp quantitative nonembeddability of the Heisenberg group into superreflexive Banach spaces

Details Sharp quantitative nonembeddability of the Heisenberg group into superreflexive Banach spaces Sharp quantitative nonembeddability of the Heisenberg group into superreflexive Banach spaces

2010-07-24

arxiv.org/abs/1007.4238v1

Mathematics

Metric Geometry

Scientific paper

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.

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