Compression bounds for Lipschitz maps from the Heisenberg group to $L_1$

We prove a quantitative bi-Lipschitz nonembedding theorem for the Heisenberg group with its Carnot-Carath'eodory metric and apply it to give a lower bound on the integrality gap of the Goemans-Linial semidefinite relaxation of the Sparsest Cut problem.