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Dynamics of \(\text{Out}(F_n)\) on the boundary of outer space. (English) Zbl 1045.20034
The outer automorphism group \(\text{Out}(F_n)\) of the free group \(F_n\) of rank \(n\) acts properly discontinuously on outer space \(CV_n\) which is a contractible space introduced by M. Culler and K. Vogtman in analogy with Teichmüller space and the action of the mapping class or modular group [Invent. Math. 84, 91-119 (1986; Zbl 0589.20022)]; it is the set of minimal free isometric actions of \(F_n\) on simplicial \(\mathbb{R}\)-trees modulo equivariant homothety and has a natural compactification in the set of minimal isometric actions of \(F_n\) on \(\mathbb{R}\)-trees.
“In this paper, we study the dynamics of the action of \(\text{Out}(F_n)\) on the boundary \(\partial CV_n\) of outer space: we describe a proper closed \(\text{Out}(F_n)\)-invariant subset \({\mathcal F}_n\) of \(\partial CV_n\) such that \(\text{Out}(F_n)\) acts properly discontinuously on the complementary open set. Moreover, we prove that there is precisely one minimal non-empty closed invariant subset \({\mathcal M}_n\) in \({\mathcal F}_n\). This set \({\mathcal M}_n\) is the closure of the \(\text{Out}(F_n)\)-orbit of any simplicial action lying in \({\mathcal F}_n\). We also prove that \({\mathcal M}_n\) contains every action having at most \(n-1\) ergodic measures. This makes us suspect that \({\mathcal M}_n={\mathcal F}_n\). Thus \({\mathcal F}_n\) would be the limit set of \(\text{Out}(F_n)\), the complement of \({\mathcal F}_n\) being its set of discontinuity.”

MSC:
20F65 Geometric group theory
20E08 Groups acting on trees
57M07 Topological methods in group theory
37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
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