A hyperbolic \(\text{Out}(F_n)\)-complex.

*(English)*Zbl 1190.20017The very short abstract of this very interesting paper is very comprehensive: “For any finite collection \(f_i\) of fully irreducible automorphisms of the free group \(F_n\) we construct a connected \(\delta\)-hyperbolic \(\text{Out}(F_n)\)-complex in which each \(f_i\) has positive translation length.”

However the statement of the main theorem gives the essence of the paper: For any finite collection \(f_1,\dots,f_k\) of fully irreducible elements of \(\text{Out}(F_n)\) there is a connected \(\delta\)-hyperbolic graph \(\mathcal X\) equipped with an (isometric) action of \(\text{Out}(F_n)\) such that:

However the statement of the main theorem gives the essence of the paper: For any finite collection \(f_1,\dots,f_k\) of fully irreducible elements of \(\text{Out}(F_n)\) there is a connected \(\delta\)-hyperbolic graph \(\mathcal X\) equipped with an (isometric) action of \(\text{Out}(F_n)\) such that:

- \(\bullet\)
- the stabilizer in \(\text{Out}(F_n)\) of a simplicial tree in \(\overline{\mathcal{PT}}\) has bounded orbits,
- \(\bullet\)
- the stabilizer in \(\text{Out}(F_n)\) of a proper free factor \(F\subset F_n\) has bounded orbits, and
- \(\bullet\)
- \(f_1,\dots,f_k\) have nonzero translation lengths.

Reviewer: Stylianos Andreadakis (Athens)

##### MSC:

20E05 | Free nonabelian groups |

20F28 | Automorphism groups of groups |

20E36 | Automorphisms of infinite groups |

20F65 | Geometric group theory |

57M07 | Topological methods in group theory |

##### Keywords:

fully irreducible automorphisms; free groups; connected hyperbolic complexes; translation lengths; hyperbolic graphs; isometric actions; outer space; measured geodesic currents##### References:

[1] | Y. Algom-Kfir, Strongly contracting geodesics in Outer space. Preprint 2008. · arxiv.org |

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