Existence and generic pff decompositions of fully singular representatives for ageometric fully irreducible outer automorphisms

Establish that every ageometric fully irreducible outer automorphism in Out(F_r) admits a fully singular train track representative, and determine whether such fully singular representatives generically admit proper full fold (pff) decompositions.

Background

The paper develops train track automata for classes of fully irreducible elements of Out(F_r), focusing on proper full fold (pff) decompositions of fully singular train track representatives. A fully singular train track representative is PNP-free and has all vertices principal, while a pff decomposition is a Stallings fold decomposition using only proper full folds (and concluding with an edge-permutation isomorphism).

The authors conjecture a broad structural picture: that every ageometric fully irreducible outer automorphism possesses such a fully singular representative and, moreover, that these representatives typically have pff decompositions. This conjecture would justify concentrating on pff automata as a robust framework for generic behavior in this setting.