Geodesic axes in the pants complex of the five-holed sphere

We study the synthetic geometry of the pants graph of the 5-holed sphere, establishing the existence of geodesics connecting any vertex or ideal point to any ideal point. We prove the existence of geodesic axes for sufficiently high powers of any pseudo-Anosov mapping class, and that large link hierarchies from Harveyʼs curve graph all induce geodesic paths.