  The free splitting graph of a free group $F_n$ with $n\geq 2$ generators is a
hyperbolic ${\rm Out}(F_n)$-graph which has a geometric realization as a sphere
graph in the connected sum of $n$ copies of $S^1\times S^2$. 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 $n\geq 3$ a new hyperbolic ${\rm Out}(F_n)$-graph. If $n=3$, then
every exponentially growing element acts on this graph with positive
translation length.
