Ideal Whitehead graphs in $$\textit{Out}(F_r)$$ Out ( F r ) II: the complete graph in each rank

Abstract

We show how to construct, for each r ≥ 3, an ageometric, fully irreducible φ ∈ Out(F r ) whose ideal Whitehead graph is the complete graph on 2r − 1 vertices. This paper is the second in a series of three where we show that precisely eighteen of the twenty-one connected, simplicial, five-vertex graphs are ideal Whitehead graphs of fully irreducible φ ∈ Out(F 3 ). The result is a first step to an Out(F r ) version of the Masur-Smillie theorem proving precisely which index lists arise from singular measured foliations for pseudo-Anosov mapping classes.In this paper we additionally give a method for finding periodic Nielsen paths and prove a criterion for identifying representatives of ageometric, fully irreducible φ ∈ Out(F r ) arXiv:1301.6645v2 [math.GR]

Ideal Whitehead graphs in Out(Fr) III: Achieved graphs in rank 3

“…We proved in [Pfa13a] that in each rank r, the complete (2r − 1)-vertex graph is the ideal Whitehead graph for some fully irreducible φ ∈ Out(F r ).…”

“…This obstacle is then simple to address by taking an adequate power to fix the periodic directions. One can check for the existence of pNP's (by hand, as in [Pfa13a], or using the computer package of Thierry Coulbois, for example). While this was not necessary for constructing any of the maps appearing in the proof of Theorem 5.1, one could construct a "Nielsen path prevention sequence" (as in [Pfa13a]) to ensure that there are no pNP's.…”

“…In light of Lemma 4.5, we show here an example of how to check that all of G is "built" (we iteratively take the image under each Dg k of the edges "created" thus far): The "construction" of such a path by a construction automorphism is proved to always occur in [Pfa13a] (see Lemma 4.12 below).…”

“…While (ss2) is technical, it is very important because the purple subgraph of an ltt structure changes with a switch and so, after applying a sequence of switches, the purple edge that would have determined the next switch may no longer exist in the ltt structure. "Switch paths" are handled more formally in [Pfa13a] than we need here, as we can use the ideas behind them to suggest the construction of a sequence, that we can then check by hand is an admissible switch sequence. We describe the idea behind the paths here:…”

A train track directed random walk on Out(Fr)

“…In the Out(F r ) setting, not only is the ideal Whitehead graph IW(ϕ) a finer invariant (c.f. [Pfa13a,Pfa13b]), but it provides further information about the behavior of lamination leaves at a singularity. It is again an invariant of the conjugacy class of ϕ, also invariant under taking positive powers of ϕ.…”