Applications and Generalizations
- The first major application of train tracks was given in the original 1992 paper of Bestvina and Handel where train tracks were introduced. The paper gave a proof of 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, is either reducible, of finite order or pseudo-anosov.
- Train tracks are the main tool in Los' algorithm for deciding whether or not two irreducible elements of Out(Fn) are conjugate in Out(Fn).
- 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 Levitt and Lustig showing that a fully irreducible automorphism of a Fn has "north-south" dynamics when acting on the Thurston-type compactification of the Culler–Vogtmann Outer space.
- A theorem of Bridson and Groves that for every automorphism α of Fn the mapping torus group of α satisfies a quadratic isoperimetric inequality.
- The proof by Bestvina, Feighn and Handel that the group Out(Fn) satisfies the Tits alternative.
- An algorithm that, given an automorphism α of Fn, decides whether or not the fixed subgroup of α is trivial and finds a finite generating set for that fixed subgroup.
- The proof of algorithmic solvability of the conjugacy problem for free-by-cyclic groups by Bogopolski, Martino, Maslakova, and Ventura.
- The machinery of train tracks for injective endomorphisms of free groups, generalizing the case of automorphisms, was developed in a 1996 book of Dicks and Ventura.
Read more about this topic: Train Track Map