 
Summary: AUTOMORPHISMS OF THE GRAPH OF FREE
SPLITTINGS
JAVIER ARAMAYONA & JUAN SOUTO
Abstract. We prove that every simplicial automorphism of the
free splitting graph of a free group Fn is induced by an outer au
tomorphism 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 ver
tices are nontrivial 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 1
skeleton 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 CullerVogtmann space. We prove that the latter is invariant
