Separating Pants Decompositions in the Pants Complex

TL;DR

This paper investigates the structure of pants decompositions on surfaces by analyzing their associated graphs, providing asymptotic estimates for the maximum distance to decompositions with separating curves, especially highlighting growth rates for closed surfaces.

Contribution

It introduces a graph-based perspective on pants decompositions and derives asymptotically sharp bounds for distances in the pants complex related to surface topology.

Findings

Maximum distance grows like log(g) for genus g surfaces.

Provides a new graph-theoretic approach to analyze pants decompositions.

Establishes asymptotic bounds for distances in the pants complex.

Abstract

We study the topological types of pants decompositions of a surface by associating to any pants decomposition $P,$ in a natural way its pants decomposition graph, $\Gamma(P).$ 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 non-trivial separating curve for all surfaces of finite type. In the main theorem we provide an asymptotically sharp approximation of this non-trivial 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 $\log(g).$