New York Journal of Mathematics
Volume 21 (2015) 417-463


Catherine Pfaff

Ideal Whitehead graphs in Out(Fr). I. Some unachieved graphs

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Published: June 20, 2015
Keywords: Outer automorphism, free group, fully irreducible, ideal Whitehead graph
Subject: 20F65

Masur and Smillie, 1993, proved precisely which singularity index lists arise from pseudo-Anosov mapping classes. In search of an analogous theorem for outer automorphisms of free groups, Handel and Mosher, 2011, ask: Is each connected, simplicial, (2r-1)-vertex graph the ideal Whitehead graph of a fully irreducible ϕ ∈ Out(Fr)? We answer this question in the negative by exhibiting, for each r, examples of connected (2r-1)-vertex graphs that are not the ideal Whitehead graph of any fully irreducible ϕ ∈ Out(Fr). In the course of our proof we also develop machinery used in Pfaff, 2012, to fully answer the question in the rank-three case.

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Department of Mathematics, University of California, Santa Barbara, CA 93106-3080