The Nielsen-Thurston classification and automorphisms of a free group I

Abstract

Using train tracks and invariant laminations, Thurston has developed a whole theory to understand the dynamics and geometry of diffeomorphisms of surfaces ([Th], [Ca-Bl]). By introducing a combinatorial analogue of train tracks M. Bestvina and M. Handel [Be-Ha] have managed to analyze irreducible automorphisms of a free group, and using this analysis to bound the rank of the fixed subgroup of an automorphism by the rank of the ambient group, which was known before as the Scott conjecture. In [Se1] we introduced a dynamical-algebraic commutative diagram associated with a (limit) action of the ambient free group Fn on some real tree. This commutative diagram allows one to interpret dynamical invariants of the limit action in terms of algebraic properties of the automorphism in question and vice versa. Using this diagram we showed how to obtain the Nielsen-Thurston classification of automorphisms of surfaces on the algebraic level and some generalized versions of the Scott conjecture. In this paper we make an extensive use of the dynamical-algebraic commutative diagram and the classification of stable actions of finitely presented groups on real trees ([Ri],[Be-Fe1]) to construct a (canonical) hierarchical decomposition of a free group associated with an automorphism of it, which is, from our point of view, the analogue of the Nielsen-Thurston classification for automorphisms of free groups. It is also our belief that our hierarchical decomposition should serve as a complementary to the Bestvina-Handel train tracks and invariant laminations in studying the dynamics and combinatorics of general (rather than irreducible) automorphisms of a free group.