Ideal Whitehead graphs in Out(Fr) III: Achieved graphs in rank 3

By proving precisely which singularity index lists arise from the pair of invariant foliations for a pseudo-Anosov surface homeomorphism, Masur and Smillie [24] determined a Teichmuller flow invariant stratification of the space of quadratic differentials. In this final paper of a three-paper series, we give a first step to an Out(F-r) analog of the Masur-Smillie theorem. Since the ideal Whitehead graphs defined by Handel and Mosher [16] give a strictly finer invariant in the analogous Out(F-r) setting, we determine which of the 21 connected, simplicial, five-vertex graphs are ideal Whitehead graphs of fully irreducible outer automorphisms in Out(F-3). The rank 2 case is actually a direct consequence of the work of Masur and Smillie, as all elements of Out(F-2) are induced by surface homeomorphisms and the index list and ideal Whitehead graph for a surface homeomorphism give precisely the same data.