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

Abstract

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 Teichmüller 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 [Formula: see text] 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 [Formula: see text] setting, we determine which of the 21 connected, simplicial, five-vertex graphs are ideal Whitehead graphs of fully irreducible outer automorphisms in [Formula: see text]. The rank 2 case is actually a direct consequence of the work of Masur and Smillie, as all elements of [Formula: see text] are induced by surface homeomorphisms and the index list and ideal Whitehead graph for a surface homeomorphism give precisely the same data.