Let A = (aij) ∊ Mn(ℝ) be an n by n symmetric stochastic matrix. For p ∊ [1, ∞) and a metric space (X, dX), let γ(A, dpx) be the infimum over those γ ∊ (0,∞] for which every x1, . . . , xn ∊ X satisfy [...] Thus γ (A, dpx) measures the magnitude of the nonlinear spectral gap of the matrix A with respect to the kernel dpX : X × X →[0,∞). We study pairs of metric spaces (X, dX) and (Y, dY ) for which there exists Ψ: (0,∞)→(0,∞) such that γ (A, dpX) ≤Ψ (A, dpY ) for every symmetric stochastic A ∊ Mn(ℝ) with (A, dpY ) < ∞. When Ψ is linear a complete geometric characterization is obtained. Our estimates on nonlinear spectral gaps yield new embeddability results as well as new nonembeddability results. For example, it is shown that if n ∊ ℕ and p ∊ (2,∞) then for every f1, . . . , fn ∊ Lp there exist x1, . . . , xn ∊ L2 such that [...] and [...] This statement is impossible for p ∊ [1, 2), and the asymptotic dependence on p in (0.1) is sharp. We also obtain the best known lower bound on the Lp distortion of Ramanujan graphs, improving over the work of Matoušek. Links to Bourgain-Milman-Wolfson type and a conjectural nonlinear Maurey-Pisier theorem are studied.