Conjugacy and Centralizers for Iwip Automorphisms of Free Groups

  • Martin Lustig
Part of the Trends in Mathematics book series (TM)


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 ≤ kN − 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 fh = 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.


Initial Segment Length Function Initial Vertex Train Track Quotient Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Birkhäuser Verlag Basel/Switzerland 2007

Authors and Affiliations

  • Martin Lustig
    • 1
  1. 1.Mathématiques (LATP)Université Paul Cézanne — Aix-Marseille IIIMarseille 20France

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