Train Track Automata for Outer Automorphisms of Free Groups and Geodesics in Outer Space

The outer automorphism group of the free group Out(F_r) acts as the isometry group on the deformation space of weighted graphs, Culler-Vogtmann Outer space CV_r. The train track theory of Bestvina-Feighn-Handel bridges studying topological representatives of the group elements and geodesics in this space it acts on. We use the asymptotic conjugacy class invariant of the Handel-Mosher ideal Whitehead graph to “stratify” the space of geodesics, and the dynamically minimal “fully irreducible” outer automorphisms, into train track automata for different ideal Whitehead graphs. We then also contextualize this work in the broader program of understanding the geodesic flow. While the flow in the closed hyperbolic manifold and Teichmuller space settings is ergodic, it is unclear whether graphs live in such a nice setting. We explain some of our indicators of certain properties the flow may have. Some results presented are joint with some combinations of Y. Algom-Kfir, D. Gagnier, I. Kapovich, J. Maher, L. Mosher, and S.J. Taylor.