Submanifold projections and hyperbolicity in Out(Fₙ)

The free splitting graph of a free group Fₙ with n≥ 2 generators is a hyperbolic Out(Fₙ)-graph which has a geometric realization as a sphere graph in the connected sum of n copies of S¹× S². We use this realization to construct submanifold projections of the free splitting graph into the free splitting graphs of proper free factors. This is used to construct for n≥ 3 a new hyperbolic Out(Fₙ)-graph. If $n=3$, then every exponentially growing element acts on this graph with positive translation length.