### Upper bound for distance in the pants graph

Geometry and Topology Seminar

8th March 2022, 2:00 pm – 3:00 pm

Fry Building, 2.04

A pants decomposition of a compact orientable surface S is a maximal collection of disjoint non-parallel simple closed curves that cut S into pairs of pants. The pants graph of S is an infinite graph whose vertices are pants decompositions of S, and where two pants decompositions are connected by an edge if they differ by a certain move that exchanges exactly one curve in the pants decomposition. One motivation for studying this graph is a celebrated result of Brock stating that the pants graph is quasi-isometric to the Teichmuller space equipped with the Weil-Petersson metric. Given two pants decompositions, we give an upper bound for their distance in the pants graph as a polynomial function of the Euler characteristic of S and the logarithm of their intersection number. The proof relies on using triangulations, train tracks, and a robust algorithm of Agol, Hass, and Thurston. This is joint work with Marc Lackenby.

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