Let phi be an outer automorphism of a free group. A topological representative of phi is a marked graph G along with a homotopy equivalence f: G → G which induces the outer automorphism phi on the fundamental group of G. For any given outer automorphism, the choice of topological representative is far from unique. Handel and Bestvina showed that sufficiently nice automorphisms admit a special type of topological representative called a train track map, whose dynamics can be well understood.
In this talk I will outline the definition and motivation for train tracks, and give a sketch of Handel and Bestvina’s algorithm for finding them.
- Junior Topology and Group Theory Seminar