Bounds for the number of moves between pants decompositions, and between triangulations

Given two pants decompositions of a compact orientable surface [Formula: see text], we give an upper bound for their distance in the pants graph that depends logarithmically on their intersection number and polynomially on the Euler characteristic of [Formula: see text]. As a consequence, we find an upper bound on the volume of the convex core of a maximal cusp (which is a hyperbolic structure on [Formula: see text] where given pants decompositions of the conformal boundary are pinched to annular cusps). As a further application, we give an upper bound for the Weil–Petersson distance between two points in the Teichmüller space of [Formula: see text] in terms of their corresponding short pants decompositions. Similarly, given two one-vertex triangulations of [Formula: see text], we give an upper bound for the number of flips and twist maps needed to convert one triangulation into the other. The proofs rely on using pre-triangulations, train tracks, and an algorithm of Agol, Hass, and Thurston.