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Algorithmic detectability of iwip automorphisms. (English) Zbl 1319.20030
From the introduction: The notion of a pseudo-Anosov homeomorphism of a compact surface plays a fundamental role in low-dimensional topology and the study of mapping class groups. In the context of $$\mathrm{Out}(F_N)$$, the concept of being pseudo-Anosov has several (nonequivalent) analogs.
The first is the notion of an ‘atoroidal’ automorphism. An element $$\varphi\in\mathrm{Out}(F_N)$$ is called atoroidal if there do not exist $$m\geq 1$$, $$h\in F_N$$, $$h\neq 1$$ such that $$\varphi^m$$ preserves the conjugacy class $$[h]$$ of $$h$$ in $$F_N$$. Another, more important, free group analog of being pseudo-Anosov is the notion of a ‘fully irreducible’ or ‘iwip’ automorphism. An element $$\varphi\in\mathrm{Out}(F_N)$$ is called reducible if there exists a free product decomposition $$F_N=A_1*\cdots*A_k*C$$ with $$k\geq 1$$, $$A_i\neq 1$$ and $$A_i\neq F_N$$ such that $$\varphi$$ permutes the conjugacy classes $$[A_1],\ldots,[A_k]$$. An element $$\varphi\in\mathrm{Out}(F_N)$$ is irreducible if it is not reducible. An element $$\varphi\in\mathrm{Out}(F_N)$$ is fully irreducible or iwip (which stands for ‘irreducible with irreducible powers’) if $$\varphi^m$$ is irreducible for all integers $$m\geq 1$$ (equivalently, for all nonzero integers $$m$$). Thus, $$\varphi$$ is an iwip if and only if there do not exist a proper free factor $$A$$ of $$F_N$$ and $$m\geq 1$$ such that $$\varphi^m([A])=[A]$$.
There is no obvious approach to algorithmically deciding whether an element$$\varphi\in\mathrm{Out}(F_N)$$ is an iwip. In this note, we provide such an algorithm.
Theorem A. There exists an algorithm that, given $$N\geq 2$$ and $$\varphi\in\mathrm{Out}(F_N)$$, decides whether or not $$\varphi$$ is an iwip.

##### MSC:
 20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects) 20E36 Automorphisms of infinite groups 20E05 Free nonabelian groups
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