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Rarity of merging outer automorphisms and prevalence of pff decompositions

Ascertain whether merging outer automorphisms—ageometric fully irreducible elements of Out(F_r) for which no fully singular train track representative admits a pff decomposition—are rare; in particular, prove that most fully singular train track representatives admit proper full fold decompositions.

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Background

The authors define a phenomenon of false singularities arising in Stallings fold decompositions that include tripod folds, which temporarily create vanishing vertices not in the image of any vertex map. They call an ageometric fully irreducible outer automorphism merging if none of its fully singular train track representatives has a pff decomposition (i.e., some merging phenomenon is unavoidable).

They conjecture that such merging cases are exceptional, with the typical situation being that fully singular representatives do admit pff decompositions. Confirming this would support focusing on pff automata to capture generic dynamics and combinatorics of axes.

References

We conjecture that merging outer automorphisms are rare, and particularly that most fully singular tt representative have a pff decomposition.

— Out($F_r$) train track automata I: Proper full fold decompositions (2409.05599 - Pfaff, 9 Sep 2024) in Section “False singularities and Pff decomposition among multiple Stallings fold decompositions,” Subsection “Merging outer automorphisms and false singularities”