Large flats in the pants graph

TL;DR

This paper investigates the geometry of the pants graph associated with a surface, proving that certain subgraphs are totally geodesic and establishing the existence of large flats within the graph.

Contribution

It demonstrates that subgraphs corresponding to multicurves with simple complements are totally geodesic and identifies the presence of maximal size flats in the pants graph.

Findings

Subgraphs P(Q) are totally geodesic in P.

Existence of maximal size flats in the pants graph.

Explicit construction of large flats.

Abstract

This note is about the geometry of the pants graph P(S), a natural simplicial graph associated to a finite type topological surface S where vertices represents pants decompositions. The main result in this note ascserts that for a multicurve Q whose complement is a number of subsurfaces of complexity at most 1. We prove that the corresponding subgraph P(Q) is totally geodesic in P, previously considering this as a metric space assigning length one to each edge. A flat is a graph isomorphic to the Cayley graph of an abelian torsion free group of finite rank. As a consequence of the main theorem we make explicit the existence of maximal size flats (large flats) in the pants graph.