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Train tracks and automorphisms of free groups. (English) Zbl 0757.57004
Let $$G$$ be a graph whose fundamental group has been identified with the free group $$F_ n$$ and let $$f:G\to G$$ be a homotopy equivalence. Then $$f$$ is said to be a train track map if $$f$$ maps vertices to vertices and the restriction of $$f^ k$$ to the interior of every edge of $$G$$ is locally injective for all $$k>0$$. An outer automorphism $$A$$ of $$F_ n$$ is reducible if there are proper free factors $$F_ 1,\dots,F_ k$$ of $$F_ n$$ with $$F_ 1*\cdots*F_ k$$ a free factor of $$F_ n$$ and such that $$A$$ transitively permutes the conjugacy classes of the $$F_ i$$’s. Otherwise $$A$$ is irreducible.
The authors give a constructive proof of the following conjecture of Thurston: Every irreducible outer automorphism of $$F_ n$$ is topologically represented by a train track map.
The idea of the proof is to change a topological representative of $$A$$ by Stallings-folds, tightenings, subdivisions, and collapsing of forests. As an application it is shown that if $$\varphi:F_ n\to F_ n$$ is an automorphism in an irreducible automorphism class, then $$\text{Rank (Fix} \varphi)\leq 1$$.
The authors then define stable relative train track maps and obtain the main result of the paper: Every outer automorphism $$A$$ of $$F_ n$$ can be represented by a stable relative train track map $$f:G\to G$$. As an application the following conjecture of Scott is proved: For any automorphism $$\varphi:F_ n\to F_ n$$, $$\text{Rank(Fix}(\varphi))\leq n$$.

##### MSC:
 57M07 Topological methods in group theory 20E05 Free nonabelian groups 20F34 Fundamental groups and their automorphisms (group-theoretic aspects) 20E08 Groups acting on trees 20F65 Geometric group theory
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