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Index deficit bounds for the stable axis bundle dimension

Determine whether the index deficit ID(φ) of a fully irreducible outer automorphism φ in Out(F_r) bounds the dimension of the stable axis bundle associated to φ, and, if so, derive explicit bounds in terms of ID(φ).

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Background

The paper defines the index deficit ID(φ)=i(φ)+r−1 and relates it to the directional surplus DS(g) of certain train track representatives (Proposition: ID(φ)=DS(g)/2 in the fully singular case). Since DS(g) reflects the number of fold choices in Stallings fold decompositions, the authors speculate that ID(φ) controls geometry along axes in Outer space.

The stable axis bundle (as discussed in prior work cited as [loneaxes]) captures the local family of axes associated to a fully irreducible element. Establishing quantitative bounds of its dimension in terms of ID(φ) would connect combinatorial train track data with geometric properties of axes.

References

As DS(g) also relates to the number of choices of folds in a Stallings fold decomposition, one can conjecture that ID() would give bounds on the dimension of the stable axis bundle.

— Out($F_r$) train track automata I: Proper full fold decompositions (2409.05599 - Pfaff, 9 Sep 2024) in Remark following Proposition “Index deficit for fully preprincipal tt representatives” in Section “Fully singular outer automorphisms and train track representatives”