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