Ideal Whitehead Graphs in $Out(F_r)$ IV: Building ideal Whitehead graphs in higher ranks and ideal Whitehead graphs with cut vertices

TL;DR

This paper constructs examples of ageometric fully irreducible outer automorphisms with ideal Whitehead graphs having cut vertices across all ranks, revealing complex behaviors and expanding the known variety of such graphs.

Contribution

It introduces a method to build ideal Whitehead graphs with cut vertices in all ranks, demonstrating new structural properties of outer automorphisms.

Findings

Existence of ageometric fully irreducible automorphisms with cut vertices in ideal Whitehead graphs in all ranks

Examples of non-single-axis Handel-Mosher axis bundles in each rank

Construction of a new class of ideal Whitehead graphs via developed tools

Abstract

We provide an example in each rank of an ageometric fully irreducible outer automorphism whose ideal Whitehead graph has a cut vertex. Consequently, we show that there exist examples in each rank of Handel-Mosher axis bundles that are not just a single axis, as well as of "nongeneric" behavior in the sense of the "train track directed" random walk of Kapovich-Pfaff. The tools developed here also allow one to construct a whole new array of ideal Whitehead graphs achieved by ageometric fully irreducible outer automorphisms in all ranks.