For the pants graph, there is little known about the behaviour of geodesics, as opposed to quasi-geodesics. Brock-Masur-Minsky showed that geodesics or geodesic segments connecting endpoints satisfying a bounded combinatorics condition, such as the stable/unstable laminations of a pseudo-Anosov, all have bounded combinatorics, outside of annuli. In this paper it is shown that there exist geodesics that also have bounded combinatorics within annuli. These geodesics are shown to have finiteness properties analogous to those of tight geodesics in the complex of curves, from which rationality of stable lengths of pseudo-Anosovs acting on the pants graph then follows from the arguments of Bowditch for the curve complex.

This work was funded by a MOE AcRF-Tier 2 WBS grant Number R-146-000-143-112.