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

Journal article

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). As a further application, we give an upper bound for the Weil–Petersson distance between two points in the Teichm¨uller space of S in terms of their corresponding short pants decompositions. 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.

Authors: Lackenby, M; Yazdi, M

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: World Scientific Publishing
• Journal: Journal of Topology and Analysis
• Publication date: 2026-01-19
• Acceptance date: 2025-10-28
• DOI: 10.1142/S1793525326500056
• EISSN: 1793-7167
• ISSN: 1793-5253
• Language: English
• Keywords: mapping class group; pants decomposition; pants graph; one-vertex triangulation; spine; volumes of hyperbolic 3-manifolds; Teichmüller space; Weil-Petersson metric