Ideal Whitehead Graphs in Out(F_r) I: Some Unachieved Graphs

TL;DR

This paper investigates which graphs can be ideal Whitehead graphs of fully irreducible outer automorphisms of free groups, providing counterexamples and developing new methods for classification.

Contribution

It demonstrates that not all connected (2r-1)-vertex graphs are ideal Whitehead graphs of such automorphisms, answering a key open question negatively.

Findings

Counterexamples for all ranks r showing some graphs are not ideal Whitehead graphs.

Development of machinery for classifying fully irreducible outer automorphisms.

Complete answer to the question in rank three case.

Abstract

H. Masur and J. Smillie proved precisely which singularity index lists arise from pseudo-Anosov mapping classes. In search of an analogous theorem for outer automorphisms of free groups, Handel and Mosher ask: Is each connected, simplicial, (2r-1)-vertex graph the ideal Whitehead graph of a fully irreducible outer automorphism in Out(F_r)? We answer this question in the negative by exhibiting, for each r, examples of connected (2r-1)-vertex graphs that are not the ideal Whitehead graph of any fully irreducible outer automorphism in Out(F_r)? In the course of our proof we also develop machinery used in "Constructing and Classifying Fully Irreducible Outer Automorphisms of Free Groups" to fully answer the question in the rank-three case.