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An algorithm to detect full irreducibility by bounding the volume of periodic free factors. (English) Zbl 1336.20037
Let $$F$$ be a finitely generated free group of rank at least 2 and $$\mathrm{Out}(F)$$ the group of its outer automorphisms. An outer automorphism $$\varphi$$ of $$F$$ is called irreducible with irreducible powers (iwip), or fully irreducible, if there does not exist a proper free factor $$A$$ of $$F$$ whose conjugacy class $$[A]$$ satisfies $$\varphi^p([A])=[A]$$ for any $$p>0$$. If $$\varphi^p([A])=[A]$$ for some proper free factor $$A<F$$ and for some $$p>0$$, then it is said that $$[A]$$ (and improperly that $$A$$) is $$\varphi$$-periodic.
Fully irreducible elements are considered analogous to pseudo-Anosov mapping classes of hyperbolic surfaces. In [T. Koberda and J. Mangahas, J. Topol. Anal. 7, No. 1, 1-21 (2015; Zbl 1328.57022)] it is provided an elementary algorithm for determining whether or not a given class is pseudo-Anosov, using a method of “list and check”. Here, the authors provide, in essence, a method of “list and check” for elements of $$\mathrm{Out}(F)$$, akin to that in the above mentioned article. That is, they provide an algorithm that, given an element $$\varphi$$ expressed as a product of generators from a finite generating set of $$\mathrm{Out}(F)$$, produces a finite list of conjugacy classes of proper free factors and checks each for $$\varphi$$-periodicity. The algorithm effectively determines whether or not the given element $$\varphi$$ is fully irreducible. The length of this list is controlled by the word length of $$\varphi$$.
For this it is proved the Theorem: There is a computable constant $$C=C(\mathcal{X,S})$$ such that, for any $$\varphi\in\mathrm{Out}(F)$$, either (i) $$\varphi$$ is fully irreducible, or (ii) $$\varphi$$ has a periodic free factor of rank 1, or (iii) $$\varphi$$ has a periodic proper free factor $$A$$ such that $$\| A\|_{\mathcal X}\leq C^{|\varphi|_{\mathcal S}}$$.
Here $$\mathcal X$$ is a fixed basis of $$F$$, $$\mathcal S$$ is a fixed generating set for $$\mathrm{Out}(F)$$, $$|\varphi|_{\mathcal S}$$ denotes the word length of $$\varphi\in\mathrm{Out}(F)$$ with respect to $$\mathcal S$$ and $$\| A\|_{\mathcal X}$$ is the volume of $$A$$, namely the number of edges in the Stallings core of the graph $$T_{\mathcal X}/F$$, where $$T_{\mathcal X}$$ is the Cayley graph of $$F$$ with respect to $$\mathcal X$$.
It is noted by the authors that previously [in I. Kapovich, Bull. Lond. Math. Soc. 46, No. 2, 279-290 (2014; Zbl 1319.20030)] it is given an (other) algorithm for determining whether a given $$\varphi\in\mathrm{Out}(F)$$ is fully irreducible.

##### MSC:
 20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects) 20E36 Automorphisms of infinite groups 20E05 Free nonabelian groups 20F65 Geometric group theory 57M07 Topological methods in group theory
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