Catherine Pfaff

Constructing and classifying fully irreducible outer automorphisms of free groups

The main theorem of this document emulates, in the context of Out(F_r) theory, a mapping class group theorem (by H. Masur and J. Smillie) that determines precisely which index lists arise from pseudo-Anosov mapping classes. Since the ideal Whitehead graph gives a finer invariant in the analogous setting of a fully irreducible outer automorphism, we instead focus on determining which of the 21 connected, loop-free, 5-vertex graphs are ideal Whitehead graphs of ageometric, fully irreducible outer automorphisms of the free group of rank 3. Our main theorem accomplishes this by showing that there are precisely 18 graphs arising as such. We also give a method for identifying certain complications called periodic Nielsen paths, prove the existence of conveniently decomposed representatives of ageometric, fully irreducible outer automorphisms having connected, loop-free, (2r-1)-vertex ideal Whitehead graphs, and prove a criterion for identifying representatives of ageometric, fully irreducible outer automorphisms. The methods we use for constructing fully irreducible outer automorphisms of free groups, as well as our identification and decomposition techniques, can be used to extend our main theorem, as they are valid in any rank. Our methods of proof rely primarily on Bestvina-Feighn-Handel train track theory and the theory of attracting laminations.

Out(F 3) Index Realization

By proving precisely which singularity index lists arise from the pair of invariant foliations for a pseudo-Anosov surface homeomorphism, Masur and Smillie determined a Teichmueller flow invariant stratification of the space of quadratic differentials. In this paper we determine an analog to the theorem for

Lone axes in outer space

Out(F_3)

Ideal Whitehead Graphs in Out(F_r) I: Some Unachieved Graphs

. That is, we determine which index lists permitted by the Gaboriau-Jaeger-Levitt-Lustig index sum inequality are achieved by fully irreducible outer automorphisms of the rank-

Stable Strata of Geodesics in Outer Space

3

Normalizers and centralizers of cyclic subgroups generated by lone axis fully irreducible outer automorphisms

free group.