This map sends vertices to vertices and edges to nontrivial edge-paths with the property that for every edge e of the graph and for every positive integer n the path fn(e) is immersed, that is fn(e) is locally injective on e. Train-track maps are a key tool in analyzing the dynamics of automorphisms of finitely generated free groups and in the study of the Culler–Vogtmann Outer space.
Train track maps for free group automorphisms were introduced in a 1992 paper of Bestvina and Handel.
[1] The notion was motivated by Thurston's train tracks on surfaces, but the free group case is substantially different and more complicated.
In the same paper they introduced the notion of a relative train track and applied train track methods to solve[1] 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[2] 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 were a key tool in the proof by Bestvina, Feighn and Handel that the group Out(Fn) satisfies the Tits alternative.
A turn is an unordered pair e, h of oriented edges of Γ (not necessarily distinct) having a common initial vertex.
The following result was obtained by Bestvina and Handel in their 1992 paper[1] where train track maps were originally introduced: Let φ ∈ Out(Fk) be irreducible.
One then defines a number of different moves on topological representatives of φ that are all seen to either decrease or preserve the Perron–Frobenius eigenvalue of the transition matrix.
This guarantees that the process terminates in a finite number of steps and the last term fN of the sequence is a train track representative of φ.
A consequence (requiring additional arguments) of the above theorem is the following:[1] Unlike for elements of mapping class groups, for an irreducible φ ∈ Out(Fk) it is often the case [12] that