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The pants graph of a compact orientable surface S, defined by Hatcher and Thurston, is a simplicial graph associated with S. Given two pants decompositions of a compact orientable surface S, 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 S. As a consequence, we find an upper bound on the volume of the convex core of a maximal cusp (which is a hyperbolic structures on S ×R where given pants decompositions of the conformal boundary are pinched to annular cusps). Similarly, given two one-vertex triangulations of S, 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.

This is joint work with Marc Lackenby.