Minimal-stretch outer automorphisms and few-fold Stallings decompositions

Determine whether, for outer automorphisms of the free group F_r, the elements with minimal stretch factors admit train track representatives whose Stallings fold decompositions contain few folds.

Background

The paper studies minimal stretch factors of fully irreducible elements in Out(F_r) and observes that the specific example achieving the minimal realized stretch factor in rank 3 is represented by a train track map whose Stallings fold decomposition consists of a single fold followed by a homeomorphism.

Motivated by the observation that folds increase the number of edges in the image of a train track map, the authors suggest a potential relationship between having a minimal stretch factor and admitting a train track representative with a small number of folds, hinting at a structural optimization principle connecting dynamical complexity (stretch factor) and combinatorial complexity (fold count).