Conjugacy and Centralizers for Iwip Automorphisms of Free Groups

Abstract

An automorphism α of a free group F N of finite rank N ≥ 2 is called iwip if no positive power of α maps any proper free factor F k of F N (1 ≤ k ≤ N − 1) to a conjugate of itself. Such automorphisms have many properties analogous to pseudo-Anosov mapping classes on surfaces. In particular, Bestvina-Handel have shown that any such α is represented by a train track map f: Γ → Γ of a graph Γ with π 1Γ ≅= F N.The goal of this paper is to give a new solution of the conjugacy problem for (outer) iwip automorphisms. We show that two train track maps f: Γ → Γ and f′: Γ′ → Γ′ represent iwip automorphisms that are conjugate in Out(F N) if and only if there exists a map h: Γ# → Γ′ which satisfies f″h = hf #, where f #: Γ# → Γ# and f″: Γ″ → Γ″ are train track maps derived algorithmically from f: Γ → Γ and f′: Γ′ → Γ″ respectively, such that they represent the same pair of automorphisms. The map h maps vertices to vertices and edges to edge paths of bounded length, where the bound is derived algorithmically from f and f′.The main ingredient of the proof, a lifting theorem of certain F Nequivariant edge-isometric maps i : \( \widetilde\Gamma \) → T, where T denotes the forward limit ℝ-tree defined by α, is a strong and useful tool in other circumstances as well.