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 \frac{\|f(x)-f(y)\|}{d_W(x,y)}\lesssim \bigg(\frac{\log\log d_W(x,y)}{\log d_W(x,y)}\bigg)^{1/p}. \qquad (1) 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 \left(\frac{\log R}{\log\log R}\right)^{1/p}. \qquad (2) 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.

Locations

• HAL (Le Centre pour la Communication Scientifique Directe) - View
• arXiv (Cornell University) - View - PDF
• Groups Geometry and Dynamics - View - PDF