Train Tracks on Graphs of Groups and Outer Automorphisms of Hyperbolic Groups

Abstract

Stallings remarked that an outer automorphism of a free group may be thought of as a subdivision of a graph followed by a sequence of folds. In this thesis, we prove that automorphisms of fundamental groups of graphs of groups satisfying this condition may be represented by irreducible train track maps in the sense of Bestvina-Handel (we allow collapsing invariant subgraphs). Of course, we construct relative train track maps as well. Along the way, we give a new exposition of the Bass-Serre theory of groups acting on trees, morphisms of graphs of groups, and foldings thereof. We produce normal forms for automorphisms of free products and extend an argument of Qing-Rafi to show that they are not quasi-geodesic. As an application, we answer affirmatively a question of Paulin: outer automorphisms of finitely-generated word hyperbolic groups satisfy a dynamical trichotomy generalizing the Nielsen-Thurston "periodic, reducible or pseudo-Anosov." At the end of the thesis we collect some open problems we find interesting.