The asymptotic geometry of grafting rays for rational laminations

In this study, we investigate the extremal length of simple closed curves and develop a sharp asymptotic estimate for the extremal length of a simple closed curve along a grafting ray determined by a rational lamination. Using the estimate of the extremal length for a rational lamination, we determine the endpoint of the grafting ray on the Gardiner–Masur boundary. Moreover, we derive an explicit formula for the limiting distance between two distinct grafting rays in the Teichmüller space with the Teichmüller metric.