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

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.