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A hyperbolic \(\text{Out}(F_n)\)-complex. (English) Zbl 1190.20017
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:
the stabilizer in \(\text{Out}(F_n)\) of a simplicial tree in \(\overline{\mathcal{PT}}\) has bounded orbits,
the stabilizer in \(\text{Out}(F_n)\) of a proper free factor \(F\subset F_n\) has bounded orbits, and
\(f_1,\dots,f_k\) have nonzero translation lengths.
Here \(\overline{\mathcal{PT}}\) denotes the compactified outer space.

20E05 Free nonabelian groups
20F28 Automorphism groups of groups
20E36 Automorphisms of infinite groups
20F65 Geometric group theory
57M07 Topological methods in group theory
Full Text: DOI Link arXiv
[1] Y. Algom-Kfir, Strongly contracting geodesics in Outer space. Preprint 2008. · arxiv.org
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