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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:
\(\bullet\)
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.

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