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Irreducible outer automorphisms of a free group. (English) Zbl 0717.20026
A map of finite graphs f: \(G\to G\) will be required to take vertices to vertices and map each edge to a path in G. If G is connected and if f is a homotopy equivalence with \(f| E\) an immersion for each edge E of G, then f is said to be a topological realization of the outer automorphism \(\theta\) (f) of the fundamental group \(\pi_ 1(G,x)\) (for \(x\in V\), V the vertex set of G) determined by \(\alpha \mapsto p\circ f(\alpha)\circ p^{-1}\); here p is a path from x to f(x) and \(\alpha\) is a loop based at x. An outer automorphism \(\theta\) of a finitely generated free group is said to be reducible [see M. Bestvina, D. Handel, Train tracks and automorphisms of free groups (to appear)], if \(\theta =\theta (f)\), where f: \(G\to G\) is a topological realization, where f and G satisfy certain specific conditions. If \(\theta\) is not reducible, then it is called irreducible. An outer automorphism \(\theta\) of a free group \(F_ n\) of rank n determines an automorphism \(\theta_{ab}\) of the abelianization \(F_ n/F_ n'\cong {\mathbb{Z}}^ n\). For a basis of \({\mathbb{Z}}^ n\) the automorphism \(\theta_{ab}\) gives a matrix with characteristic polynomial denoted by char \(\theta\). Also a polynomial p(x)\(\in {\mathbb{Z}}[x]\) is called PV-polynomial if p(x) has precisely one root \(\lambda\) with \(| \lambda | >1\) and the other roots \(\mu\), \(| \mu | <1.\)
In this paper the authors using a “beautiful” result of Bestvina-Handel (loc. cit.) prove that: (i) if char \(\theta\) is a PV-polynomial, then \(\theta^ k\), \(k\geq 1\) is irreducible (ii) if char \(\theta\) is irreducible over \({\mathbb{Q}}\) and if some matrix representation for \(\theta_{ab}\) is primitive, then, for all \(k\geq 1\), \(\theta^ k\) is irreducible. (iii) If \(\phi\) is an automorphism of a finitely generated free group of rank \(\geq 3\) such that char \(\phi\) is a PV-polynomial, then Fix \(\phi\) \(=\{1\}\). (This last result settles a conjecture of Stallings.) Here a square matrix M over \({\mathbb{R}}\) is called primitive if each entry of M is nonnegative and there exists \(n\geq 1\) such that every entry of \(M^ n\) is positive.
Reviewer: S.Andreadakis

20F28 Automorphism groups of groups
57M05 Fundamental group, presentations, free differential calculus
20E05 Free nonabelian groups
20F34 Fundamental groups and their automorphisms (group-theoretic aspects)
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
Full Text: DOI
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[5] John R. Stallings, Topologically unrealizable automorphisms of free groups, Proc. Amer. Math. Soc. 84 (1982), no. 1, 21 – 24. · Zbl 0477.20012
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