Products of Farey graphs are totally geodesic in the pants graph

TL;DR

This paper proves that certain subgraphs of the pants graph, determined by fixed collections of curves, are totally geodesic, confirming a special case of a conjecture and linking Farey graph products to geodesic properties.

Contribution

It establishes that subgraphs defined by fixed curves are totally geodesic in the pants graph, resolving a specific case of a conjecture and connecting Farey graph products to geodesic structures.

Findings

Subgraphs determined by fixed curves are totally geodesic.

Embedded products of Farey graphs are totally geodesic.

A pants graph contains a convex n-flat if and only if it contains an n-quasi-flat.

Abstract

We show that for a surface S, the subgraph of the pants graph determined by fixing a collection of curves that cut S into pairs of pants, once-punctured tori, and four-times-punctured spheres is totally geodesic. The main theorem resolves a special case of a conjecture made by Aramayona, Parlier, and Shackleton and has the implication that an embedded product of Farey graphs in any pants graph is totally geodesic. In addition, we show that a pants graph contains a convex n-flat if and only if it contains an n-quasi-flat.