Stable Lengths on the pants graph are rational

TL;DR

This paper proves that stable lengths of pseudo-Anosov elements on the pants graph are rational by demonstrating the existence of geodesics with bounded combinatorics within annuli, extending known finiteness properties.

Contribution

It introduces geodesics with bounded combinatorics inside annuli in the pants graph, leading to rationality results for stable lengths of pseudo-Anosovs.

Findings

Existence of geodesics with bounded combinatorics within annuli.

Finiteness properties analogous to tight geodesics in the curve complex.

Rationality of stable lengths for pseudo-Anosovs on the pants graph.

Abstract

For the pants graph, there is little known about the behaviour of geodesics, as opposed to quasigeodesics. 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, \textit{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.