Ideal Whitehead Graphs in Out(F_r) III: Achieved Graphs in Rank 3

TL;DR

This paper classifies which specific ideal Whitehead graphs can occur for fully irreducible outer automorphisms in rank 3, advancing the understanding of $Out(F_r)$ dynamics analogous to Teichmüller theory.

Contribution

It identifies the possible ideal Whitehead graphs in rank 3, providing a key step towards an $Out(F_r)$ analog of the Masur-Smillie theorem.

Findings

Outlines which of the 21 graphs are realized as ideal Whitehead graphs in $Out(F_3)$

Establishes a classification of ideal Whitehead graphs for fully irreducible automorphisms

Advances the understanding of $Out(F_r)$ invariant structures 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 determined a Teichm\"uller 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 give a strictly finer invariant in the analogous $Out(F_r)$ setting, we determine which of the twenty-one connected, simplicial, five-vertex graphs are ideal Whitehead graphs of fully irreducible outer automorphisms in $Out(F_3)$.