an:05658118
Zbl 1190.20017
Bestvina, Mladen; Feighn, Mark
A hyperbolic \(\text{Out}(F_n)\)-complex
EN
Groups Geom. Dyn. 4, No. 1, 31-58 (2010).
00256917
2010
j
20E05 20F28 20E36 20F65 57M07
fully irreducible automorphisms; free groups; connected hyperbolic complexes; translation lengths; hyperbolic graphs; isometric actions; outer space; measured geodesic currents
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:
{\parindent=6mm
\begin{itemize}\item[{\(\bullet\)}]the stabilizer in \(\text{Out}(F_n)\) of a simplicial tree in \(\overline{\mathcal{PT}}\) has bounded orbits,
\item[{\(\bullet\)}]the stabilizer in \(\text{Out}(F_n)\) of a proper free factor \(F\subset F_n\) has bounded orbits, and
\item[{\(\bullet\)}]\(f_1,\dots,f_k\) have nonzero translation lengths.
\end{itemize}}
Here \(\overline{\mathcal{PT}}\) denotes the compactified outer space.
Stylianos Andreadakis (Athens)