Automorphisms of the Graph of Free Splittings

We prove that every simplicial automorphism of the free splitting graph of a free group Fn is induced by an outer automorphism of Fn for n ≥ 3. In this note we consider the graph Gn of free splittings of the free group Fn of rank n ≥ 3. Loosely speaking, Gn is the graph whose vertices are non-trivial free splittings of Fn up to conjugacy, and where two vertices are adjacent if they are represented by free splittings admitting a common refinement. The group Out(Fn) of outer automorphisms of Fn acts simplicially on Gn. Denoting by Aut(Gn) the group of simplicial automorphisms of the free splitting graph, we prove: Theorem 1. The natural map Out(Fn) → Aut(Gn) is an isomorphism for n ≥ 3. We briefly sketch the proof of Theorem 1. We identify Gn with the 1skeleton of the sphere complex Sn and observe that every automorphism of Gn extends uniquely to an automorphism of Sn. It is due to Hatcher [4] that the sphere complex contains an embedded copy of the spine Kn of Culler-Vogtmann space. We prove that the latter is invariant under Aut(Sn), and that the restriction homomorphism Aut(Sn) → Aut(Kn) is injective. The claim of Theorem 1 then follows from a result of Bridson-Vogtmann [1] which asserts that Out(Fn) is the full automorphism group of Kn. We are grateful to our motherland for the beauty of its villages.