arXiv:2611.14566v1 Announce Type: new Abstract: We study automorphism orbit graphs of oriented conjugacy classes in the free group of rank two, with every elementary Whitehead move assigned unit cost. Collapsing signed permutations of the basis turns each orbit component into a tree of edges and triangles. We obtain this description by combining the known rank-two conjugacy-class stabilizer structure with the Bass-Serre tree of the rank-two outer automorphism group. The simplified graph and all its vertex-induced subgraphs have optimal thin-triangle constant zero for tree components and 3/4 otherwise. Every whole orbit component of the original graph has thin-triangle constant at most 2, while every component of every vertex-induced subgraph has constant at most 5/2. Both bounds are sharp in the primitive orbit and apply to the full geometric realization, including edge-interior triangle corners. The hereditary upper estimate also follows from known chordality theory; the whole-component improvement uses the geometry of adjacent permutation fibres. These results complement earlier fixed-length and minimum-length orbit classifications.
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