Uniform thickness along the connectivity path for reducibility detection

Establish whether the points of Culler–Vogtmann outer space encountered along the connectivity path used to detect reducibility can be guaranteed to remain uniformly thick.

Background

The reducibility-detection algorithm connects a uniform rose representing the input basis to a point having an invariant core subgraph, using the connectivity theorem for displacement level sets. The proof controls the algorithm while the path remains in the thick part of Culler–Vogtmann space, then uses the first point leaving that region to produce a visibly reducible representative.

The authors explicitly state that they do not know whether all points encountered on the relevant path remain uniformly thick. This unresolved issue is why the proof must select the maximal initial thick segment and separately analyze the first thin point.

References

Proof of Theorem 1.8: We proceed much as in the proof of Theorem 1.4, but here we do not know that the points in CVn we encounter will remain uniformly thick.

— On the connectivity of level sets of automorphisms of free groups, with applications to decision problems  (1703.09945 - Francaviglia et al., 2017) in Proof of Theorem 1.8, Section 11, p. 59