A "cubist" decomposition of the Handel-Mosher axis bundle
Catherine Eva Pfaff, Chi Cheuk Tsang
Published: 2025/3/20
Abstract
We show that the axis bundle of a nongeometric fully irreducible outer automorphism admits a canonical "cubist" decomposition into branched cubes that fit together with special combinatorics. From this structure, we locate a canonical finite collection of periodic fold lines in each axis bundle. This can be considered as an analogue of results of Hamenst\"adt and Agol from the surface setting, which state that the set of trivalent train tracks carrying the unstable lamination of a pseudo-Anosov map can be given the structure of a CAT(0) cube complex, and that there is a canonical periodic fold line in this cube complex. This work also gives a "hands on" solution to the fully irreducible conjugacy problem in $\mathrm{Out}(F_r)$, and an answer to questions of Handel-Mosher and Bridson-Vogtmann regarding the geometry of the axis bundle.