The distribution of critical graphs ofJenkins–Strebel differentials

By work of Jenkins and Strebel, given a Riemann surface X and a simple closed multicurve on it, there exists a unique quadratic differential q on X whose horizontal foliation is measure equivalent to .We study the distribution of the critical graphs of these differentials in the moduli space of metric ribbon graphs as the extremal length of the multicurves goes to infinity, showing they equidistribute to the Kontsevich measure regardless of the initial choice of X. 32G15, 57K20 1. Introduction 2571 2. Preliminaries 2576 3. Near the multiple zero locus 2581 4. Uniform geometric estimates 2585 5. Horoball measures 2589 6. Equidistribution of critical graphs 2597