Vertical versus horizontal Poincaré inequalities on the Heisenberg group

Let ℍ = 〈a, b|a[a, b] = [a, b]a ∧ b[a, b] = [a, b]b〉 be the discrete Heisenberg group, equipped with the left-invariant word metric d_W(·, ·) associated to the generating set {a, b, a⁻¹, b⁻¹}. Letting B_n = {x ∈ ℍ: d_W(x, e_ℍ) ⩽ n} denote the corresponding closed ball of radius n ∈ ℕ, and writing c = [a, b] = aba⁻¹b⁻¹, we prove that if (X, ‖ · ‖X) is a Banach space whose modulus of uniform convexity has power type q ∈ [2,∞), then there exists K ∈ (0, ∞) such that every f: ℍ → X satisfies (Formula Presented). It follows that for every n ∈ ℕ the bi-Lipschitz distortion of every f: B_n → X is at least a constant multiple of (log n)^1/q, an asymptotically optimal estimate as n → ∞.