Sharp quantitative nonembeddability of the Heisenberg group into superreflexive Banach spaces

Let H denote the discrete Heisenberg group, equipped with a word metric dW associated to some finite symmetric generating set. We show that if (X, ‖ ·‖) is a p-convex Banach space then for any Lipschitz function f : H→ X there exist x, y ∈ H with dW (x, y) arbitrarily large and ‖f(x)− f(y)‖ dW (x, y) . ( log log dW (x, y) log dW (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 ( logR log logR )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.