History
Train track maps for free group automorphisms were introduced in a 1992 paper of Bestvina and Handel. The notion was motivated by Thurston's train tracks on surfaces, but the free group case is substantially different and more complicated. In their 1992 paper Bestvina and Handel proved that every irreducible automorphism of Fn has a train-track representative. In the same paper they introduced the notion of a relative train track and applied train track methods to solve the Scott conjecture which says that for every automorphism α of a finitely generated free group Fn the fixed subgroup of α is free of rank at most n. In a subsequent paper Bestvina and Handel applied the train track techniques to obtain an effective proof of Thurston's classification of homeomorphisms of compact surfaces (with or without boundary) which says that every such homeomorphism is, up to isotopy, either reducible, of finite order or pseudo-anosov.
Since then train tracks became a standard tool in the study of algebraic, geometric and dynamical properties of automorphisms of free groups and of subgroups of Out(Fn). Train tracks are particularly useful since they allow to understand long-term growth (in terms of length) and cancellation behavior for large iterates of an automorphism of Fn applied to a particular conjugacy class in Fn. This information is especially helpful when studying the dynamics of the action of elements of Out(Fn) on the Culler–Vogtmann Outer space and its boundary and when studying Fn actions of on real trees. Examples of applications of train tracks include: a theorem of Brinkmann proving that for an automorphism α of Fn the mapping torus group of α is word-hyperbolic if and only if α has no periodic conjugacy classes; a theorem of Bridson and Groves that for every automorphism α of Fn the mapping torus group of α satisfies a quadratic isoperimetric inequality; a proof of algorithmic solvability of the conjugacy problem for free-by-cyclic groups; and others.
Train tracks were a key tool in the proof by Bestvina, Feighn and Handel that the group Out(Fn) satisfies the Tits alternative.
The machinery of train tracks for injective endomorphisms of free groups was later developed by Dicks and Ventura.
Read more about this topic: Train Track Map
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