Abstract
We show how to construct, for each \(r \ge 3\), an ageometric, fully irreducible \(\phi \in Out(F_r)\) 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 \(\phi \in Out(F_3)\). 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 \(\phi \in Out(F_r)\).
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