Let denote the discrete Heisenberg group, equipped with a word metric associated to some finite symmetric generating set. We show that if is a -convex Banach space then for any Lipschitz function there exist with arbitrarily large and
We also show that any embedding into of a ball of radius in 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 is Hilbert space we obtain a representation-theoretic proof yielding bounds corresponding to (1) and (2) which are sharp up to a universal constant.
Cite this article
Tim Austin, Assaf Naor, Romain Tessera, Sharp quantitative nonembeddability of the Heisenberg group into superreflexive Banach spaces. Groups Geom. Dyn. 7 (2013), no. 3, pp. 497–522DOI 10.4171/GGD/193