Ideal Whitehead Graphs in Out(F_r) III: Achieved Graphs in Rank 3

Out(F3) Index Realization

“…A particularly good example of this arises when trying to generalize the Masur-Smillie pseudo-Anosov index theorem to nongeometric fully irreducibles. One facet of this depth is expanded upon in [Pfa12b], [Pfa13a], and [Pfa13b], where we show that, unlike with pseudo-Anosovs, where the ideal Whitehead graph can be determined by the singularity index list, the ideal Whitehead graph actually gives a finer invariant of a fully irreducible giving, in particular, more detailed behavior of the lamination at a singularity. In this paper we focus on the fact that, instead of being restricted by an index sum equality, such as the Poincaré-Hopf index equality, the index sum for a fully irreducible is only restricted by an inequality.…”

“…INDEX LIST (− 32 ): A plethora of examples with this index list can be found in[Pfa13b].INDEX LIST (− 1 2 , −1): The representative on the graph b → bdadb c → cdbdbdadbdbdadbdbdbdadbdbdadbcdbdbdadbdbdadb d → dbd As you can see from the below figure, the local Whitehead graphs are connected.Restricting to periodic directions, since there are no periodic Nielsen paths, this gives the ideal Whitehead graph, from which the index list is computed to be (− 1 2 , −1):…”

Lone axes in outer space

“…What we prove in Theorem 4.7 of Section 4 is a necessary and sufficient condition for an ageometric fully irreducible outer automorphism to have a unique axis. One may note that examples of fully irreducibles satisfying the conditions of Theorem 4.7 can be found in [Pfa13a] and [Pfa13b] and it was in fact proved later, in [KP14], that satisfying these conditions is generic along a particular "train track directed" random walk.…”

“…Unlike in the surface case where one has the Poincaré-Hopf index equality, Gaboriau, Jäger, Levitt, and Lustig proved in [GJLL98] that there is instead a rotationless index inequality 0 > i(ϕ) ≥ 1 − r that each fully irreducible ϕ ∈ Out(F r ) satisfies. (Here we have rewritten the inequality using the [Pfa13b] rotationless index definition, revised to be invariant under taking powers and to have its sign be consistent with the mapping class group case. )…”

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.…”