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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.