Constructing and Classifying Fully Irreducible Outer Automorphisms of Free Groups

Catherine Pfaff (Rutgers)

The main goal of my research is to prove a fully irreducible outer automorphisms analogue of the Howard Masur and John Smillie theorem which precisely records the lists of foliation singularity indices which arise from pseudo-Anosov elements of Mapping Class Groups. The more appropriate conjectured theorem in the Out(F_r) setting actually records possibilities for an even finer outer automorphism invariant, an ideal Whitehead graph. I have developed methods for constructing and identifying fully irreducible outer automorphisms that have thus far been used to completely determine which connected graphs with 5 vertices occur as ideal Whitehead graphs for ageometric fully irreducible outer automorphisms of free groups. These methods will further allow us to construct a variety of examples of a more general nature and in a higher rank and may eventually be extended to completely determine our analogue theorem. Finally, I have theorems ruling out the possibility of a category of graphs arising as ideal Whitehead graphs of ageometric fully irreducible outer automorphisms in any rank r=3.