Bestvina–Handel theorem for irreducible automorphisms

Statement

The following result was obtained by Bestvina and Handel in their 1992 paper where train track maps were originally introduced:

Let φ ∈ Out(F_k) be irreducible. Then there exists a train track representative of φ.

For a topological representative f:Γ→Γ of an automorphism φ of F_k the transition matrix M(f) is an rxr matrix (where r is the number of topological edges of Γ) where the entry m_ij is the number of times the path f(e_j) passes through the edge e_i (in either direction). If φ is irreducible, the transition matrix M(f) is irreducible in the sense of the Perron–Frobenius theorem and it has a unique Perron–Frobenius eigenvalue λ(f) ≥ 1 which is equal to the spectral radius of M(f).

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. These moves include: subdividing an edge; valence-one homotopy (getting rid of a degree-one vertex); valence-two homotopy (getting rid of a degree-two vertex); collapsing an invariant forest; and folding. Of these moves the valence-one homotopy always reduced the Perron–Frobenius eigenvalue.

Starting with some topological representative f of an irreducible automorphism φ one then algorithmically constructs a sequence of topological representatives

f = f₁, f₂, f₃,...

of φ where f_n is obtained from f_n−1 by several moves, specifically chosen. In this sequence, if f_n is not a train track map, then the moves producing f_n+1 from f_n necessarily involve a sequence of folds followed by a valence-one homotopy, so that the Perron–Frobenius eigenvalue of f_n+1 is strictly smaller than that of f_n. The process is arranged in such a way that Perron–Frobenius eigenvalues of the maps f_n take values in a discrete substet of $\mathbb R$. This guarantees that the process terminates in a finite number of steps and the last term f_N of the sequence is a train track representative of φ.

A consequence (requiring additional arguments) of the above theorem is the following:

• If φ ∈ Out(F_k) is irreducible then the Perron–Frobenius eigenvalue λ(f) does not depend on the choice of a train track representative f of φ but is uniquely determined by φ itself and is denoted by λ(φ). The number λ(φ) is called the growth rate of φ.
• If φ ∈ Out(F_k) is irreducible and of infinite order then λ(φ) > 1. Moreover, in this case for every free basis X of F_k and for most nontrivial values of w ∈ F_k there exists C ≥ 1 such that for all n ≥ 1

$$\frac{1}{C}\lambda^n(\phi) \le ||\phi^n(w)||_X\le C \lambda^n(\phi),$$

where ||u||_X is the cyclically reduced length of an element u of F_k with respect to X. The only exceptions occur when F_k corresponds to the fundamental group of a compact surface with boundary S, and φ corresponds to a pseudo-Anosov homeomorphism of S, and w corresponds to a path going around a component of the boundary of S.

Unlike for elements of mapping class groups, for an irreducible φ ∈ Out(F_k) it is often the case that

λ(φ) ≠ λ(φ⁻¹).