Constructing and Classifying Fully Irreducible Outer Automorphisms of Free Groups

TL;DR

This paper classifies which specific graphs can serve as ideal Whitehead graphs for ageometric, fully irreducible outer automorphisms of free groups of rank 3, extending understanding of their structure and construction methods.

Contribution

It precisely determines the 18 out of 21 possible ideal Whitehead graphs that correspond to such automorphisms and provides methods for their identification and construction.

Findings

18 graphs arise as ideal Whitehead graphs for automorphisms

Methods for identifying periodic Nielsen paths are developed

Construction techniques are extended to arbitrary rank

Abstract

The main theorem of this document emulates, in the context of Out(F_r) theory, a mapping class group theorem (by H. Masur and J. Smillie) that determines precisely which index lists arise from pseudo-Anosov mapping classes. Since the ideal Whitehead graph gives a finer invariant in the analogous setting of a fully irreducible outer automorphism, we instead focus on determining which of the 21 connected, loop-free, 5-vertex graphs are ideal Whitehead graphs of ageometric, fully irreducible outer automorphisms of the free group of rank 3. Our main theorem accomplishes this by showing that there are precisely 18 graphs arising as such. We also give a method for identifying certain complications called periodic Nielsen paths, prove the existence of conveniently decomposed representatives of ageometric, fully irreducible outer automorphisms having connected, loop-free, (2r-1)-vertex ideal Whitehead graphs, and prove a criterion for identifying representatives of ageometric, fully irreducible outer automorphisms. The methods we use for constructing fully irreducible outer automorphisms of free groups, as well as our identification and decomposition techniques, can be used to extend our main theorem, as they are valid in any rank. Our methods of proof rely primarily on Bestvina-Feighn-Handel train track theory and the theory of attracting laminations.