An algorithm to detect full irreducibility by bounding the volume of periodic free factors.

*(English)*Zbl 1336.20037Let \(F\) 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.

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.

Reviewer: Dimitrios Varsos (AthĂna)

##### 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 |