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; ⋯ ℍ with d_W (x, y) arbitrarily large and (eqution presented) 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 (eqution presented) 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.