Products of Farey graphs are totally geodesic in the pants graph

Abstract

We show that for a surface [Formula: see text], the subgraph of the pants graph determined by fixing a collection of curves that cut [Formula: see text] 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 in [2] 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 [Formula: see text]-flat if and only if it contains an [Formula: see text]-quasi-flat.