$Out(F_3)$ Index Realization

Abstract: By proving precisely which singularity index lists arise from the pair of invariant foliations for a pseudo-Anosov surface homeomorphism, Masur and Smillie determined a Teichmueller flow invariant stratification of the space of quadratic differentials. In this paper we determine an analog to the theorem for $Out(F_3)$. That is, we determine which index lists permitted by the Gaboriau-Jaeger-Levitt-Lustig index sum inequality are achieved by fully irreducible outer automorphisms of the rank-$3$ free group.