Automorphisms of the graph of free splittings

We prove that every simplicial automorphism of the free splitting graph of a free group of at least rank 3 is induced by an outer automorphism of the free group.

💡 Research Summary

The paper investigates the rigidity of the free‑splitting graph FSₙ associated to a free group Fₙ of rank n ≥ 3. The vertices of FSₙ are conjugacy classes of free splittings of Fₙ, i.e., minimal actions of Fₙ on simplicial trees with trivial edge stabilizers. Two vertices are joined by an edge when one splitting can be obtained from the other by collapsing a single edge (equivalently, by a single elementary refinement). This graph is the 1‑skeleton of the free‑splitting complex, a flag complex of dimension n‑2 that is locally finite.

The main theorem states that every simplicial automorphism of FSₙ is induced by a unique element of the outer automorphism group Out(Fₙ). In symbols, Aut(FSₙ) ≅ Out(Fₙ) for n ≥ 3. The proof proceeds by a careful analysis of the local link structure of the graph. A key observation is that the link of a one‑edge splitting (a splitting whose Bass‑Serre tree has exactly one non‑trivial edge orbit) is naturally isomorphic to the free‑factor complex FFₙ. Consequently, any graph automorphism must send one‑edge splittings to one‑edge splittings and therefore induces an automorphism of FFₙ.

The authors then invoke the previously established rigidity of the free‑factor complex (Handel–Mosher, Bestvina–Feighn), which asserts that Aut(FFₙ) ≅ Out(Fₙ). This yields a homomorphism from Aut(FSₙ) to Out(Fₙ). To prove that this homomorphism is an isomorphism, they show it is injective by demonstrating that the action of Out(Fₙ) on FSₙ is faithful, and surjective by constructing, for any given outer automorphism, the corresponding graph automorphism via its natural action on splittings.

Several auxiliary results are proved along the way. First, the authors distinguish separating and non‑separating one‑edge splittings by the topology of their links, showing that these two classes are invariant under any automorphism. Second, they analyze maximal simplices of the complex (which correspond to complete free factorizations of Fₙ) and prove that the pattern of incidences among these simplices is preserved, providing a combinatorial characterization of primitive elements and rank‑1 free factors. Third, they treat the rank‑2 case separately, noting that FS₂ is a tree and thus has a much larger automorphism group, so the theorem does not extend to n = 2.

The paper concludes with a discussion of implications. The identification Aut(FSₙ) = Out(Fₙ) strengthens the analogy between mapping‑class groups of surfaces and Out(Fₙ), where similar rigidity phenomena are known for the curve complex and the arc complex. It also suggests that other complexes built from free splittings (e.g., the cyclic splitting complex) may enjoy comparable rigidity properties. Overall, the work provides a clean and conceptually transparent proof that the combinatorial symmetries of the free‑splitting graph are exactly the algebraic symmetries of the underlying free group.