Sharp quantitative nonembeddability of the Heisenberg group into superreflexive Banach spaces

  title={Sharp quantitative nonembeddability of the Heisenberg group into superreflexive Banach spaces},
  author={T. Austin and A. Naor and R. Tessera},
  journal={arXiv: Metric Geometry},
  • T. Austin, A. Naor, R. Tessera
  • Published 2010
  • Mathematics
  • arXiv: Metric Geometry
  • 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… CONTINUE READING
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