Sharp quantitative nonembeddability of the Heisenberg group into superreflexive Banach spaces

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, parallel to . parallel to) is a p-convex Banach space then for any Lipschitz function f: H -> X there exist x, y is an element of H with d(W) (x, y) arbitrarily large and parallel to f(x) - f(y)parallel to/d(W)(x, y) less than or similar to (log log d(W)(x, y)/log d(W)(x, y))(1/p). (1) We also show that any embedding into X of a ball of radius R >= 4 in H incurs bi-Lipschitz distortion that grows at least as a constant multiple of (log R/log log R)(1/p). (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.