Separating pants decompositions in the pants complex

摘要

We study the topological types of pants decompositions of a surface by associating to any pants decomposition P, its pants decomposition graph, Γ(P). This perspective provides a convenient way to analyze the maximum distance in the pants complex of any pants decomposition to a pants decomposition containing a nontrivial separating curve for all surfaces of finite type. We provide an asymptotically sharp approximation of this nontrivial distance in terms of the topology of the surface. In particular, for closed surfaces of genus g we show the maximum distance in the pants complex of any pants decomposition to a pants decomposition containing a separating curve grows asymptotically like the function log(g). The lower bounds follow from an explicit constructive algorithm for an infinite family of high girth log-length connected graphs, which may be of independent interest.