Bestvina, Mladen; Feighn, Mark A hyperbolic \(\text{Out}(F_n)\)-complex. (English) Zbl 1190.20017 Groups Geom. Dyn. 4, No. 1, 31-58 (2010). The 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: {\(\bullet\)}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. Here \(\overline{\mathcal{PT}}\) denotes the compactified outer space. Reviewer: Stylianos Andreadakis (Athens) Cited in 46 Documents 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 PDFBibTeX XMLCite \textit{M. Bestvina} and \textit{M. Feighn}, Groups Geom. Dyn. 4, No. 1, 31--58 (2010; Zbl 1190.20017) Full Text: DOI arXiv Link References: [1] Y. Algom-Kfir, Strongly contracting geodesics in Outer space. Preprint 2008. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.