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 \(\varphi:F_ n\to F_ n\), \(\text{Rank(Fix}(\varphi))\leq n\).


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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