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Ideal Whitehead graphs in $$\mathrm{Out}(F_r)$$. II: The complete graph in each rank. (English) Zbl 1367.20049
Summary: We show how to construct, for each $$r \geq 3$$, an ageometric, fully irreducible $$\phi \in \mathrm{Out}(F_r)$$ 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 $$\phi \in \mathrm{Out}(F_3)$$. The result is a first step to an $$\mathrm{Out}(F_r)$$ 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 $$\phi \in \mathrm{Out}(F_r)$$.
For Part I, see [the author, New York J. Math. 21, 417–463 (2015; Zbl 1355.20036)].

##### MSC:
 20F65 Geometric group theory 05C25 Graphs and abstract algebra (groups, rings, fields, etc.) 20E05 Free nonabelian groups 20F36 Braid groups; Artin groups 20F28 Automorphism groups of groups 57M50 General geometric structures on low-dimensional manifolds
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